Any product of (even infinitely many) injective modules is injective; conversely, if a direct product of modules is injective, then each module is injective . Every direct sum of finitely many injective modules is injective. In general, submodules, factor modules, or infinite direct sums of injective modules need not be injective. Every submodule of every injective module is injective if and only if the ring is Artinian semisimple ; every factor module of every injective module is injective if and only if the ring is hereditary, ; every infinite direct sum of injective modules is injective if and only if the ring is Noetherian, .
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The injective hull of a module is the smallest injective module containing the given one and was described in . One can use injective hulls to define a minimal injective resolution (see below). If each term of the injective resolution is the injective hull of the cokernel of the previous map, then the injective resolution has minimal length.
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Injective resolutions measure how far from injective a module is in terms of the injective dimension and represent modules in the derived category. Injective hulls are maximal essential extensions, and turn out to be minimal injective extensions. Over a Noetherian ring, every injective module is uniquely a direct sum of indecomposable modules, and their structure is well understood. An injective module over one ring, may not be injective over another, but there are well- understood methods of changing rings which handle special cases.
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In mathematics, particularly in algebra, the injective hull (or injective envelope) of a module is both the smallest injective module containing it and the largest essential extension of it. Injective hulls were first described in .
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Every module M also has an injective resolution: an exact sequence of the form :0 → M → I0 → I1 → I2 → ... where the I j are injective modules. Injective resolutions can be used to define derived functors such as the Ext functor. The length of a finite injective resolution is the first index n such that In is nonzero and Ii = 0 for i greater than n. If a module M admits a finite injective resolution, the minimal length among all finite injective resolutions of M is called its injective dimension and denoted id(M).
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However, projective and injective objects in A do not necessarily correspond to projective and injective R-modules.
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7 This is a special feature of hereditary rings like the integers Z: the direct sum of injective modules is injective because the ring is Noetherian, and the quotients of injectives are injective because the ring is hereditary, so any submodule generated by injective modules is injective. The converse is a result of : if every module has a unique maximal injective submodule, then the ring is hereditary. A complete classification of countable reduced periodic abelian groups is given by Ulm's theorem.
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Every ring with unity is a free module and hence is a projective as a module over itself, but it is rarer for a ring to be injective as a module over itself, . If a ring is injective over itself as a right module, then it is called a right self-injective ring. Every Frobenius algebra is self-injective, but no integral domain that is not a field is self- injective. Every proper quotient of a Dedekind domain is self-injective.
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Rings which are themselves injective modules have a number of interesting properties and include rings such as group rings of finite groups over fields. Injective modules include divisible groups and are generalized by the notion of injective objects in category theory.
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A right Noetherian, right self-injective ring is called a quasi-Frobenius ring, and is two-sided Artinian and two-sided injective, . An important module theoretic property of quasi-Frobenius rings is that the projective modules are exactly the injective modules.
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For instance, if R is a subring of S such that S is a flat R-module, then every injective S-module is an injective R-module. In particular, if R is an integral domain and S its field of fractions, then every vector space over S is an injective R-module. Similarly, every injective R[x]-module is an injective R-module. For quotient rings R/I, the change of rings is also very clear.
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Suppose . So ⇒ ⇒ . Therefore, it follows from the definition that f is injective. There are multiple other methods of proving that a function is injective.
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If M does not admit a finite injective resolution, then by convention the injective dimension is said to be infinite. As an example, consider a module M such that id(M) = 0\. In this situation, the exactness of the sequence 0 → M → I0 → 0 indicates that the arrow in the center is an isomorphism, and hence M itself is injective.A module isomorphic to an injective module is of course injective.
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The notion of injective object in the category of abelian groups was studied somewhat independently of injective modules under the term divisible group. Here a Z-module M is injective if and only if n⋅M = M for every nonzero integer n. Here the relationships between flat modules, pure submodules, and injective modules is more clear, as it simply refers to certain divisibility properties of module elements by integers.
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In an injective space, the radius of the minimum ball that contains any set S is equal to half the diameter of S. This follows since the balls of radius half the diameter, centered at the points of S, intersect pairwise and therefore by hyperconvexity have a common intersection; a ball of radius half the diameter centered at a point of this common intersection contains all of S. Thus, injective spaces satisfy a particularly strong form of Jung's theorem. Every injective space is a complete space , and every metric map (or, equivalently, nonexpansive mapping, or short map) on a bounded injective space has a fixed point (; ). A metric space is injective if and only if it is an injective object in the category of metric spaces and metric maps. For additional properties of injective spaces see .
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Therefore, the finitely generated injective left A-modules are precisely the modules of the form Homk(P, k) where P is a finitely generated projective right A-module. For symmetric algebras, the duality is particularly well-behaved and projective modules and injective modules coincide. For any Artinian ring, just as for commutative rings, there is a 1-1 correspondence between prime ideals and indecomposable injective modules. The correspondence in this case is perhaps even simpler: a prime ideal is an annihilator of a unique simple module, and the corresponding indecomposable injective module is its injective hull.
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In Baer's original paper, he proved a useful result, usually known as Baer's Criterion, for checking whether a module is injective: a left R-module Q is injective if and only if any homomorphism g : I → Q defined on a left ideal I of R can be extended to all of R. Using this criterion, one can show that Q is an injective abelian group (i.e. an injective module over Z). More generally, an abelian group is injective if and only if it is divisible. More generally still: a module over a principal ideal domain is injective if and only if it is divisible (the case of vector spaces is an example of this theorem, as every field is a principal ideal domain and every vector space is divisible). Over a general integral domain, we still have one implication: every injective module over an integral domain is divisible.
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An R-module is an R/I-module precisely when it is annihilated by I. The submodule annI(M) = { m in M : im = 0 for all i in I } is a left submodule of the left R-module M, and is the largest submodule of M that is an R/I-module. If M is an injective left R-module, then annI(M) is an injective left R/I-module. Applying this to R=Z, I=nZ and M=Q/Z, one gets the familiar fact that Z/nZ is injective as a module over itself. While it is easy to convert injective R-modules into injective R/I-modules, this process does not convert injective R-resolutions into injective R/I-resolutions, and the homology of the resulting complex is one of the early and fundamental areas of study of relative homological algebra.
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In mathematics, Nakayama's conjecture is a conjecture about Artinian rings, introduced by . The generalized Nakayama conjecture is an extension to more general rings, introduced by . proved some cases of the generalized Nakayama conjecture. Nakayama's conjecture states that if all the modules of a minimal injective resolution of an Artin algebra R are injective and projective, then R is self-injective.
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For this purpose, one abstractly defines a field extension as an injective ring homomorphism between two fields. Every non-zero ring homomorphism between fields is injective because fields do not possess nontrivial proper ideals, so field extensions are precisely the morphisms in the category of fields. Henceforth, we will suppress the injective homomorphism and assume that we are dealing with actual subfields.
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A module E is called the injective hull of a module M, if E is an essential extension of M, and E is injective. Here, the base ring is a ring with unity, though possibly non-commutative.
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2 of Every object in a Grothendieck category has an injective hull.
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An injectively immersed submanifold that is not an embedding. If M is compact, an injective immersion is an embedding, but if M is not compact then injective immersions need not be embeddings; compare to continuous bijections versus homeomorphisms.
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There is also an algebraic version of incompressibility. Suppose \iota: S \rightarrow M is a proper embedding of a compact surface in a 3-manifold. Then S is π1-injective (or algebraically incompressible) if the induced map :\iota_\star: \pi_1(S) \rightarrow \pi_1(M) on fundamental groups is injective. In general, every π1-injective surface is incompressible, but the reverse implication is not always true.
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For instance, the Lens space L(4,1) contains an incompressible Klein bottle that is not π1-injective. However, if S is two-sided, the loop theorem implies Kneser's lemma, that if S is incompressible, then it is π1-injective.
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A ring is left coherent if and only if every direct product of flat right modules is flat , . Compare this to: A ring is left Noetherian if and only if every direct sum of injective left modules is injective.
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For finite-dimensional algebras over fields, these injective hulls are finitely-generated modules .
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1-59 The machinery of train tracks for injective endomorphisms of free groups was later developed by Dicks and Ventura.Warren Dicks, and Enric Ventura. The group fixed by a family of injective endomorphisms of a free group. Contemporary Mathematics, 195.
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In some cases, for R a subring of a self-injective ring S, the injective hull of R will also have a ring structure. For instance, taking S to be a full matrix ring over a field, and taking R to be any ring containing every matrix which is zero in all but the last column, the injective hull of the right R-module R is S. For instance, one can take R to be the ring of all upper triangular matrices. However, it is not always the case that the injective hull of a ring has a ring structure, as an example in shows. A large class of rings which do have ring structures on their injective hulls are the nonsingular rings.
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For all common algebraic structures, and, in particular for vector spaces, an injective homomorphism is also called a monomorphism. However, in the more general context of category theory, the definition of a monomorphism differs from that of an injective homomorphism. This is thus a theorem that they are equivalent for algebraic structures; see for more details. A function f that is not injective is sometimes called many- to-one.
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More generally, let C be an abelian category. An object E is an injective hull of an object M if M → E is an essential extension and E is an injective object. If C is locally small, satisfies Grothendieck's axiom AB5 and has enough injectives, then every object in C has an injective hull (these three conditions are satisfied by the category of modules over a ring).Section III.
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Formally, this follows from the fact that the code C is an injective map.
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As stated above, any abelian group A can be uniquely embedded in a divisible group D as an essential subgroup. This divisible group D is the injective envelope of A, and this concept is the injective hull in the category of abelian groups.
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Then is injective and continuous, the domain is open in , but the image is not.
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If R is a principal left ideal domain, then divisible modules coincide with injective modules. Thus in the case of the ring of integers Z, which is a principal ideal domain, a Z-module (which is exactly an abelian group) is divisible if and only if it is injective. If R is a commutative domain, then the injective R modules coincide with the divisible R modules if and only if R is a Dedekind domain.
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The various concepts relating to functions can also be generalised to binary functions. For example, the division example above is surjective (or onto) because every rational number may be expressed as a quotient of an integer and a natural number. This example is injective in each input separately, because the functions f x and f y are always injective. However, it's not injective in both variables simultaneously, because (for example) f (2,4) = f (1,2).
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The monomorphisms in Met are the injective metric maps. The epimorphisms are the metric maps for which the domain of the map has a dense image in the range. The isomorphisms are the isometries, i.e. metric maps which are injective, surjective, and distance-preserving.
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Thus one speaks of a P resolution. In particular, every module has free resolutions, projective resolutions and flat resolutions, which are left resolutions consisting, respectively of free modules, projective modules or flat modules. Similarly every module has injective resolutions, which are right resolutions consisting of injective modules.
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In mathematics, a V-ring is a ring R such that every simple R-module is injective. The following three conditions are equivalent: #Every simple left (resp. right) R-module is injective #The radical of every left (resp. right) R-module is zero #Every left (resp.
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In mathematics, especially in the area of abstract algebra known as module theory, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers. Specifically, if Q is a submodule of some other module, then it is already a direct summand of that module; also, given a submodule of a module Y, then any module homomorphism from this submodule to Q can be extended to a homomorphism from all of Y to Q. This concept is dual to that of projective modules. Injective modules were introduced in and are discussed in some detail in the textbook . Injective modules have been heavily studied, and a variety of additional notions are defined in terms of them: Injective cogenerators are injective modules that faithfully represent the entire category of modules.
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In mathematics, especially in the field of category theory, the concept of injective object is a generalization of the concept of injective module. This concept is important in cohomology, in homotopy theory and in the theory of model categories. The dual notion is that of a projective object.
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In mathematics, algebraically compact modules, also called pure-injective modules, are modules that have a certain "nice" property which allows the solution of infinite systems of equations in the module by finitary means. The solutions to these systems allow the extension of certain kinds of module homomorphisms. These algebraically compact modules are analogous to injective modules, where one can extend all module homomorphisms. All injective modules are algebraically compact, and the analogy between the two is made quite precise by a category embedding.
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Similarly, an integral domain is a Dedekind domain if and only if every divisible module over it is injective.
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In particular, for an integral domain, the injective hull of the ring (considered as a module over itself) is the field of fractions. The injective hulls of nonsingular rings provide an analogue of the ring of quotients for non-commutative rings, where the absence of the Ore condition may impede the formation of the classical ring of quotients. This type of "ring of quotients" (as these more general "fields of fractions" are called) was pioneered in , and the connection to injective hulls was recognized in .
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Manhattan metric, has a connected orthogonal convex hull, then that hull coincides with the tight span of the points. In metric geometry, the metric envelope or tight span of a metric space M is an injective metric space into which M can be embedded. In some sense it consists of all points "between" the points of M, analogous to the convex hull of a point set in a Euclidean space. The tight span is also sometimes known as the injective envelope or hyperconvex hull of M. It has also been called the injective hull, but should not be confused with the injective hull of a module in algebra, a concept with a similar description relative to the category of R-modules rather than metric spaces.
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It is not true in general, however, that all monomorphisms must be injective in other categories; that is, there are settings in which the morphisms are functions between sets, but one can have a function that is not injective and yet is a monomorphism in the categorical sense. For example, in the category Div of divisible (abelian) groups and group homomorphisms between them there are monomorphisms that are not injective: consider, for example, the quotient map , where Q is the rationals under addition, Z the integers (also considered a group under addition), and Q/Z is the corresponding quotient group. This is not an injective map, as for example every integer is mapped to 0. Nevertheless, it is a monomorphism in this category.
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T-colorings correspond to homomorphisms into certain infinite graphs. An oriented coloring of a directed graph is a homomorphism into any oriented graph. An L(2,1)-coloring is a homomorphism into the complement of the path graph that is locally injective, meaning it is required to be injective on the neighbourhood of every vertex.
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One can create a bit-commitment scheme from any one-way function that is injective. The scheme relies on the fact that every one-way function can be modified (via the Goldreich-Levin theorem) to possess a computationally hard-core predicate (while retaining the injective property). Let f be an injective one-way function, with h a hard-core predicate. Then to commit to a bit b Alice picks a random input x and sends the triple :(h,f(x),b \oplus h(x)) to Bob, where \oplus denotes XOR, i.e.
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A Jordan curve or a simple closed curve in the plane R2 is the image C of an injective continuous map of a circle into the plane, φ: S1 → R2. A Jordan arc in the plane is the image of an injective continuous map of a closed and bounded interval into the plane. It is a plane curve that is not necessarily smooth nor algebraic. Alternatively, a Jordan curve is the image of a continuous map φ: [0,1] → R2 such that φ(0) = φ(1) and the restriction of φ to [0,1) is injective.
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In an injective rule, every two different configurations have different successors, but Rule 90 has pairs of configurations with the same successor. Rule 90 provides an example of a cellular automaton that is surjective but not injective. The Garden of Eden theorem of Moore and Myhill implies that every injective cellular automaton must be surjective, but this example shows that the converse is not true.. , pp. 959–960. provide a similar analysis of the predecessors of the same rule for finite sets of cells with periodic boundary conditions.
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The empty metric space is the initial object of Met; any singleton metric space is a terminal object. Because the initial object and the terminal objects differ, there are no zero objects in Met. The injective objects in Met are called injective metric spaces. Injective metric spaces were introduced and studied first by , prior to the study of Met as a category; they may also be defined intrinsically in terms of a Helly property of their metric balls, and because of this alternative definition Aronszajn and Panitchpakdi named these spaces hyperconvex spaces.
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If there is a surjection from to that is not injective, then no surjection from to is injective. In fact no function of any kind from to is injective. This is not true for infinite sets: Consider the function on the natural numbers that sends 1 and 2 to 1, 3 and 4 to 2, 5 and 6 to 3, and so on. There is a similar principle for infinite sets: If uncountably many pigeons are stuffed into countably many pigeonholes, there will exist at least one pigeonhole having uncountably many pigeons stuffed into it.
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A surjunctive group is a group with the property that, when its elements are used as the cells of cellular automata, every injective transition function of a cellular automaton is also surjective. Equivalently, summarizing the definitions above, a group G is surjunctive if, for every finite set S, every continuous equivariant injective function f:S^G \to S^G is also surjective.Ceccherini-Silberstein & Coornaert (2010) p.57 The implication from injectivity to surjectivity is a form of the Garden of Eden theorem, and the cellular automata defined from injective and surjective transition functions are reversible.
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Such transformation turns the given Euclidean structure into a (isomorphic but) different Euclidean structure; both Euclidean structures correspond to a single topological structure. In contrast, the transition from "3-dim Euclidean" to "Euclidean" is not forgetful; a Euclidean space need not be 3-dimensional, but if it happens to be 3-dimensional, it is full-fledged, no structure is lost. In other words, the latter transition is injective (one-to- one), while the former transition is not injective (many-to-one). We denote injective transitions by an arrow with a barbed tail, "↣" rather than "→".
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In the language of the category theory, an injective homomorphism is also called a monomorphism and a surjective homomorphism an epimorphism.
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In mathematics, the horizontal line test is a test used to determine whether a function is injective (i.e., one-to-one).
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If the homomorphism from R to the new ring is to be injective, no further elements can be given an inverse.
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The mapping of Z into X is in fact an embedding. This is a consequence of being quasi-invertible in if and only if it is quasi-invertible in . Indeed, if is injective on A, its restriction to is also injective. Conversely, the two equations for the quasi- inverse in imply that it is also a quasi-inverse in .
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Any locally convex space having the extension property is injective. If is an injective Banach space, then for every Banach space , every continuous linear operator from a vector subspace of into has a continuous linear extension to all of . In 1953, Alexander Grothendieck that any Banach space with the extension property is either finite-dimensional or else not separable.
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It is possible to restate the characteristic property of a cancellative element in terms of a property held by the corresponding left multiplication and right multiplication maps defined by and . An element a in S is left cancellative if and only if La is injective. An element a is right cancellative if and only if Ra is injective.
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If is injective, the subalgebra generated by is isomorphic to . In this case, this subalgebra is often denoted by . The notation ambiguity is generally harmless, because of the isomorphism. If the evaluation homorphism is not injective, this means that its kernel is a nonzero ideal, consisting of all polynomials that become zero when is substituted for .
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An injective endomorphism can be extended to an automorphism of a magma extension--the colimit of the constant sequence of the endomorphism.
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The injective- surjective-bijective terminology (both as nouns and adjectives) was originally coined by the French Bourbaki group, before their widespread adoption.
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If every horizontal line intersects the curve of f(x) in at most one point, then f is injective or one-to-one.
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Any metric space has a smallest injective metric space into which it can be isometrically embedded, called its metric envelope or tight span.
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These groups are particularly interesting because they are the only known examples of Noetherian group rings , or group rings of finite injective dimension.
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In fact, to turn an injective function into a bijective (hence invertible) function, it suffices to replace its codomain Y by its actual range . That is, let such that for all x in X; then g is bijective. Indeed, f can be factored as , where is the inclusion function from J into Y. More generally, injective partial functions are called partial bijections.
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Reshetnyak's theorem implies that all pure topological results about analytic functions (such that the Maximum Modulus Principle, Rouché's theorem etc.) extend to quasiregular maps. Injective quasiregular maps are called quasiconformal. A simple example of non-injective quasiregular map is given in cylindrical coordinates in 3-space by the formula : (r,\theta,z)\mapsto (r,2\theta,z). This map is 2-quasiregular.
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In general, for an algebraic category C, an embedding between two C-algebraic structures X and Y is a C-morphism that is injective.
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This shows that the retraction is impossible, because again the retraction would induce an injective group homomorphism from the latter to the former group.
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In mathematics, the injective tensor product of two topological vector spaces was introduced by Alexander Grothendieck and was used by him to define nuclear spaces.
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Equivalently, the injective dimension of M is the minimal integer (if there is such, otherwise ∞) n such that Ext(–,M) = 0 for all N > n.
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If X is itself injective, then we can choose the injective resolution 0 → X → X → 0, and we obtain that RiF(X) = 0 for all i ≥ 1. In practice, this fact, together with the long exact sequence property, is often used to compute the values of right derived functors. An equivalent way to compute RiF(X) is the following: take an injective resolution of X as above, and let Ki be the image of the map Ii-1→Ii (for i=0, define Ii-1=0), which is the same as the kernel of Ii→Ii+1. Let φi : Ii-1→Ki be the corresponding surjective map.
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Trivially, the zero module {0} is injective. Given a field k, every k-vector space Q is an injective k-module. Reason: if Q is a subspace of V, we can find a basis of Q and extend it to a basis of V. The new extending basis vectors span a subspace K of V and V is the internal direct sum of Q and K. Note that the direct complement K of Q is not uniquely determined by Q, and likewise the extending map h in the above definition is typically not unique. The rationals Q (with addition) form an injective abelian group (i.e.
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In algebraic geometry, a morphism of schemes :f: X -> Y is called radicial or universally injective, if, for every field K the induced map X(K) → Y(K) is injective. (EGA I, (3.5.4)) This is a generalization of the notion of a purely inseparable extension of fields (sometimes called a radicial extension, which should not be confused with a radical extension.) It suffices to check this for K algebraically closed. This is equivalent to the following condition: f is injective on the topological spaces and for every point x in X, the extension of the residue fields :k(f(x)) ⊂ k(x) is radicial, i.e.
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The Tietze extension theorem can be used to show that an interval is an injective cogenerator in a category of topological spaces subject to separation axioms.
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In mathematics, specifically category theory, an essential monomorphism is a monomorphism f in a category C such that for a morphism g in C, g \circ f is a monomorphism only when g is a monomorphism. Essential monomorphisms in a category of modules are those whose image is an essential submodule of the codomain. An injective hull of an object X is an essential monomorphism from X to an injective object.
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Since S in the construction contains no zero divisors, the natural map R \to Q(R) is injective, so the total quotient ring is an extension of R.
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Several distinct definitions generalize divisible groups to divisible modules. The following definitions have been used in the literature to define a divisible module M over a ring R: # rM = M for all nonzero r in R. (It is sometimes required that r is not a zero-divisor, and some authors require that R is a domain.) # For every principal left ideal Ra, any homomorphism from Ra into M extends to a homomorphism from R into M. (This type of divisible module is also called principally injective module.) # For every finitely generated left ideal L of R, any homomorphism from L into M extends to a homomorphism from R into M. The last two conditions are "restricted versions" of the Baer's criterion for injective modules. Since injective left modules extend homomorphisms from all left ideals to R, injective modules are clearly divisible in sense 2 and 3. If R is additionally a domain then all three definitions coincide.
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This class turns out to be distinct from both the surjective and injective automata, and in some subsequent research, automata with this property have been called invertible finite automata.
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An R module M has finite uniform dimension (=finite rank) n if and only if the injective hull of M is a finite direct sum of n indecomposable submodules.
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It is important to be able to consider modules over subrings or quotient rings, especially for instance polynomial rings. In general, this is difficult, but a number of results are known, . Let S and R be rings, and P be a left-R, right-S bimodule that is flat as a left-R module. For any injective right S-module M, the set of module homomorphisms HomS( P, M ) is an injective right R-module.
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One also talks about injective objects in categories more general than module categories, for instance in functor categories or in categories of sheaves of OX-modules over some ringed space (X,OX). The following general definition is used: an object Q of the category C is injective if for any monomorphism f : X → Y in C and any morphism g : X → Q there exists a morphism h : Y → Q with hf = g.
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Let X and Y be measurable spaces. If there exist injective, bimeasurable maps f : X \to Y, g : Y \to X, then X and Y are isomorphic (the Schröder–Bernstein property).
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252, Theorem 10.1. Usually the morphisms induced by inclusion in this theorem are not themselves injective, and the more precise version of the statement is in terms of pushouts of groups.
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In many circumstances conditions are imposed on the modules Ei resolving the given module M. For example, a free resolution of a module M is a left resolution in which all the modules Ei are free R-modules. Likewise, projective and flat resolutions are left resolutions such that all the Ei are projective and flat R-modules, respectively. Injective resolutions are right resolutions whose Ci are all injective modules. Every R-module possesses a free left resolution.
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This right adjoint sends the injective hull E(k) mentioned above to k, which is a dualizing object in D(k). This abstract fact then gives rise to the above-mentioned equivalence.
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In category theory, a faithful functor (respectively a full functor) is a functor that is injective (respectively surjective) when restricted to each set of morphisms that have a given source and target.
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The characteristic of an integral domain is either 0 or a prime number. If R is an integral domain of prime characteristic p, then the Frobenius endomorphism f(x) = xp is injective.
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Reinhold Baer (22 July 1902 – 22 October 1979) was a German mathematician, known for his work in algebra. He introduced injective modules in 1940. He is the eponym of Baer rings and Baer groups.
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In mathematics, it can be shown that every function can be written as the composite of a surjective function followed by an injective function. Factorization systems are a generalization of this situation in category theory.
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Frobenius algebras were generalized to quasi-Frobenius rings, those Noetherian rings whose right regular representation is injective. In recent times, interest has been renewed in Frobenius algebras due to connections to topological quantum field theory.
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This image of the open interval (with boundary points identified with the arrow marked ends) is an immersed submanifold. An immersed submanifold of a manifold M is the image S of an immersion map f: N → M; in general this image will not be a submanifold as a subset, and an immersion map need not even be injective (one-to-one) – it can have self-intersections.. More narrowly, one can require that the map f: N → M be an injection (one-to-one), in which we call it an injective immersion, and define an immersed submanifold to be the image subset S together with a topology and differential structure such that S is a manifold and the inclusion f is a diffeomorphism: this is just the topology on N, which in general will not agree with the subset topology: in general the subset S is not a submanifold of M, in the subset topology. Given any injective immersion f : N → M the image of N in M can be uniquely given the structure of an immersed submanifold so that f : N → f(N) is a diffeomorphism. It follows that immersed submanifolds are precisely the images of injective immersions.
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All field homomorphisms are injective. If is also surjective, it is called an isomorphism (or the fields and are called isomorphic). A field is called a prime field if it has no proper (i.e., strictly smaller) subfields.
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It is also obvious that the map is both injective and surjective; meaning that is a bijective homomorphism, i.e. an isomorphism. It follows that, as claimed: In 1847, Cauchy used this approach to define the complex numbers.
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In mathematics, a surjunctive group is a group such that every injective cellular automaton with the group elements as its cells is also surjective. Surjunctive groups were introduced by . It is unknown whether every group is surjunctive.
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A typical example of a monotone function is the following function on the cycle with 6 elements: : : : A function is called an embedding if it is both monotone and injective. Equivalently, an embedding is a function that pushes forward the ordering on : whenever , one has . As an important example, if is a subset of a cyclically ordered set , and is given its natural ordering, then the inclusion map is an embedding. Generally, an injective function from an unordered set to a cycle induces a unique cyclic order on that makes an embedding.
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T-theory originated from a question raised by Manfred Eigen in the late 1970s. He was trying to fit twenty distinct t-RNA molecules of the Escherichia coli bacterium into a tree. An important concept of T-theory is the tight span of a metric space. If X is a metric space, the tight span T(X) of X is, up to isomorphism, the unique minimal injective metric space that contains X. John Isbell was the first to discover the tight span in 1964, which he called the injective envelope.
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In mathematics, injective sheaves of abelian groups are used to construct the resolutions needed to define sheaf cohomology (and other derived functors, such as sheaf Ext). There is a further group of related concepts applied to sheaves: flabby (flasque in French), fine, soft (mou in French), acyclic. In the history of the subject they were introduced before the 1957 "Tohoku paper" of Alexander Grothendieck, which showed that the abelian category notion of injective object sufficed to found the theory. The other classes of sheaves are historically older notions.
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Related terms such as domain, codomain, injective, and continuous can be applied equally to maps and functions, with the same meaning. All these usages can be applied to "maps" as general functions or as functions with special properties.
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In the branch of abstract mathematics called category theory, a projective cover of an object X is in a sense the best approximation of X by a projective object P. Projective covers are the dual of injective envelopes.
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The space Aim(X) is injective (hyperconvex in the sense of Aronszajn-Panitchpakdi) – given a metric space M, which contains Aim(X) as a metric subspace, there is a canonical (and explicit) metric retraction of M onto Aim(X) .
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In category theory, the notion of a projective object generalizes the notion of a projective module. Projective objects in abelian categories are used in homological algebra. The dual notion of a projective object is that of an injective object.
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The construction of the dual vector space mentioned in the introduction is an example of such a duality. Indeed, the set of morphisms, i.e., linear maps, forms a vector space in its own right. The map mentioned above is always injective.
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The Dedekind–MacNeille completion is self-dual: the completion of the dual of a partial order is the same as the dual of the completion.. The Dedekind–MacNeille completion of has the same order dimension as does itself.This result is frequently attributed to an unpublished 1961 Harvard University honors thesis by K. A. Baker, "Dimension, join-independence and breadth in partially ordered sets". It was published by . In the category of partially ordered sets and monotonic functions between partially ordered sets, the complete lattices form the injective objects for order-embeddings, and the Dedekind–MacNeille completion of is the injective hull of ..
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Every morphism in a concrete category whose underlying function is injective is a monomorphism; in other words, if morphisms are actually functions between sets, then any morphism which is a one-to-one function will necessarily be a monomorphism in the categorical sense. In the category of sets the converse also holds, so the monomorphisms are exactly the injective morphisms. The converse also holds in most naturally occurring categories of algebras because of the existence of a free object on one generator. In particular, it is true in the categories of all groups, of all rings, and in any abelian category.
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The function given by is not injective, since each possible result y (except 0) corresponds to two different starting points in – one positive and one negative, and so this function is not invertible. With this type of function, it is impossible to deduce a (unique) input from its output. Such a function is called non-injective or, in some applications, information-losing. If the domain of the function is restricted to the nonnegative reals, that is, the function is redefined to be with the same rule as before, then the function is bijective and so, invertible.
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The field of fractions K of an integral domain R is the set of fractions a/b with a and b in R and b ≠ 0 modulo an appropriate equivalence relation, equipped with the usual addition and multiplication operations. It is "the smallest field containing R " in the sense that there is an injective ring homomorphism such that any injective ring homomorphism from R to a field factors through K. The field of fractions of the ring of integers \Z is the field of rational numbers \Q. The field of fractions of a field is isomorphic to the field itself.
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In general topology, an embedding is a homeomorphism onto its image.. . More explicitly, an injective continuous map f : X \to Y between topological spaces X and Y is a topological embedding if f yields a homeomorphism between X and f(X) (where f(X) carries the subspace topology inherited from Y). Intuitively then, the embedding f : X \to Y lets us treat X as a subspace of Y. Every embedding is injective and continuous. Every map that is injective, continuous and either open or closed is an embedding; however there are also embeddings which are neither open nor closed. The latter happens if the image f(X) is neither an open set nor a closed set in Y. For a given space Y, the existence of an embedding X \to Y is a topological invariant of X. This allows two spaces to be distinguished if one is able to be embedded in a space while the other is not.
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Clearly, \phi\circ\psi is the identity; in particular, φ is surjective. To see φ is injective, suppose φ(λ) = 0. Consider :\phi(\lambda)(t_1 v_1 + \cdots + t_q v_q) = \lambda(t_1 v_1 + \cdots + t_q v_q, ..., t_1 v_1 + \cdots + t_q v_q), which is zero.
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In mathematics, demonic composition is an operation on binary relations that is somewhat comparable to ordinary composition of relations but is robust to refinement of the relations into (partial) functions or injective relations. Unlike ordinary composition of relations, demonic composition is not associative.
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Minkowski space has four dimensions and indices 3 and 1 (assignment of "+" and "−" to them differs depending on conventions). Purely algebraic statements (ones that do not use positivity) usually only rely on the nondegeneracy (the injective homomorphism ) and thus hold more generally.
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Since h is a computationally hard-core predicate, recovering h(x) from f(x) with probability greater than one-half is as hard as inverting f. Perfect binding follows from the fact that f is injective and thus f(x) has exactly one preimage.
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Many common notions from mathematics (e.g. surjective, injective, free object, basis, finite representation, isomorphism) are definable purely in category theoretic terms (cf. monomorphism, epimorphism). Category theory has been suggested as a foundation for mathematics on par with set theory and type theory (cf. topos).
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A group G is called co-Hopfian if whenever \varphi:G\to G is an injective group homomorphism then \varphi is surjective, that is \varphi(G)=G.P. de la Harpe, Topics in geometric group theory. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 2000.
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In relative homological algebra, the extension property of homomorphisms may be required only for certain submodules, rather than for all. For instance, a pure injective module is a module in which a homomorphism from a pure submodule can be extended to the whole module.
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Group homomorphisms tend to reduce the orders of elements: if f: G → H is a homomorphism, and a is an element of G of finite order, then ord(f(a)) divides ord(a). If f is injective, then ord(f(a)) = ord(a). This can often be used to prove that there are no (injective) homomorphisms between two concretely given groups. (For example, there can be no nontrivial homomorphism h: S3 → Z5, because every number except zero in Z5 has order 5, which does not divide the orders 1, 2, and 3 of elements in S3.) A further consequence is that conjugate elements have the same order.
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An abelian group (G, +) is divisible if, for every positive integer n and every g \in G, there exists y \in G such that ny=g.Griffith, p.6 An equivalent condition is: for any positive integer n, nG=G, since the existence of y for every n and g implies that n G\supseteq G, and in the other direction n G\subseteq G is true for every group. A third equivalent condition is that an abelian group G is divisible if and only if G is an injective object in the category of abelian groups; for this reason, a divisible group is sometimes called an injective group.
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Every configuration of Rule 90 has exactly four predecessors, other configurations that form the given configuration after a single step. Therefore, in contrast to many other cellular automata such as Conway's Game of Life, Rule 90 has no Garden of Eden, a configuration with no predecessors. It provides an example of a cellular automaton that is surjective (each configuration has a predecessor) but not injective (it has sets of more than one configuration with the same successor). It follows from the Garden of Eden theorem that Rule 90 is locally injective (all configurations with the same successor vary at an infinite number of cells).
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The companion terms monomorphism and epimorphism were originally introduced by Nicolas Bourbaki; Bourbaki uses monomorphism as shorthand for an injective function. Early category theorists believed that the correct generalization of injectivity to the context of categories was the cancellation property given above. While this is not exactly true for monic maps, it is very close, so this has caused little trouble, unlike the case of epimorphisms. Saunders Mac Lane attempted to make a distinction between what he called monomorphisms, which were maps in a concrete category whose underlying maps of sets were injective, and monic maps, which are monomorphisms in the categorical sense of the word.
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This construction differs slightly from the one in (Macdonald, 1979). That construction only uses the surjective morphisms ρn without mentioning the injective morphisms φn: it constructs the homogeneous components of ΛR separately, and equips their direct sum with a ring structure using the ρn. It is also observed that the result can be described as an inverse limit in the category of graded rings. That description however somewhat obscures an important property typical for a direct limit of injective morphisms, namely that every individual element (symmetric function) is already faithfully represented in some object used in the limit construction, here a ring R[X1,...,Xd]Sd.
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The tensor product also operates on linear maps between vector spaces. Specifically, given two linear maps and between vector spaces, the tensor product of the two linear maps and is a linear map :S\otimes T:V\otimes W\to X\otimes Y defined by :(S\otimes T)(v\otimes w)=S(v)\otimes T(w). In this way, the tensor product becomes a bifunctor from the category of vector spaces to itself, covariant in both arguments. If and are both injective, surjective or (in the case that , , , and are normed vector spaces or topological vector spaces) continuous, then is injective, surjective or continuous, respectively.
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Topological groups for which the Bohr compactification mapping is injective are called maximally almost periodic (or MAP groups). In the case G is a locally compact connected group, MAP groups are completely characterized: They are precisely products of compact groups with vector groups of finite dimension.
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For example, a module has projective dimension zero if and only if it is a projective module. If M does not admit a finite projective resolution then the projective dimension is infinite. For example, for a commutative local ring R, the projective dimension is finite if and only if R is regular and in this case it coincides with the Krull dimension of R. Analogously, the injective dimension id(M) and flat dimension fd(M) are defined for modules also. The injective and projective dimensions are used on the category of right R modules to define a homological dimension for R called the right global dimension of R. Similarly, flat dimension is used to define weak global dimension.
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Quotients of flat modules are not in general flat. For example, for each integer n > 1, \Z/n\Z is not flat over \Z, because n: \Z \to \Z, x \mapsto nx is injective, but tensored with \Z/n\Z it is not. Similarly, \Q/\Z is not flat over \Z.
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Functions with left inverses are always injections. That is, given , if there is a function such that for every , :g(f(x)) = x (f can be undone by g), then f is injective. In this case, g is called a retraction of f. Conversely, f is called a section of g.
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A map f: A → X of topological spaces is a (Hurewicz) cofibration if it has the homotopy extension property for maps to any space. This is one of the central concepts of homotopy theory. A cofibration f is always injective, in fact a homeomorphism to its image.Hatcher (2002), Proposition 4H.1.
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In particular, any module over a semisimple ring is injective and projective. Since "projective" implies "flat", a semisimple ring is a von Neumann regular ring. Semisimple rings are of particular interest to algebraists. For example, if the base ring R is semisimple, then all R-modules would automatically be semisimple.
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In classical mathematics, every injective function with a nonempty domain necessarily has a left inverse; however, this may fail in constructive mathematics. For instance, a left inverse of the inclusion of the two-element set in the reals violates indecomposability by giving a retraction of the real line to the set .
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Since is strictly increasing it is injective, and hence a homeomorphism; and by the theorem of differentiation of the inverse function, its inverse has a finite derivative at any point, which vanishes at least at the points . These form a dense subset of (actually, it vanishes in many other points; see below).
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A more extreme example is the map defined by because here g is injective and continuous but does not even yield a homeomorphism onto its image. The theorem is also not generally true in infinite dimensions. Consider for instance the Banach space l∞ of all bounded real sequences. Define as the shift .
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A proof that a function f is injective depends on how the function is presented and what properties the function holds. For functions that are given by some formula there is a basic idea. We use the definition of injectivity, namely that if , then . Here is an example: : f = 2x + 3 Proof: Let .
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The category has a projective generator (Z) and an injective cogenerator (Q/Z). Given two abelian groups A and B, their tensor product A⊗B is defined; it is again an abelian group. With this notion of product, Ab is a closed symmetric monoidal category. Ab is not a topos since e.g.
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Alain Connes studies operator algebras. In his early work on von Neumann algebras in the 1970s, he succeeded in obtaining the almost complete classification of injective factors. He also formulated the Connes embedding problem. Following this, he made contributions in operator K-theory and index theory, which culminated in the Baum–Connes conjecture.
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This branch cut separates the principal branch from the two branches and . In all branches with , there is a branch point at and a branch cut along the entire negative real axis. The functions are all injective and their ranges are disjoint. The range of the entire multivalued function is the complex plane.
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Any semisimple algebra over the complex numbers C of finite dimension can be expressed as a direct sum ⊕k Mnk(C) of matrix algebras, and the C-algebra homomorphisms between two such algebras up to inner automorphisms on both sides are completely determined by the multiplicity number between 'matrix algebra' components. Thus, an injective homomorphism of ⊕k=1i Mnk(C) into ⊕l=1j Mml(C) may be represented by a collection of positive numbers ak, l satisfying ∑ nk ak, l ≤ ml. (The equality holds if and only if the homomorphism is unital; we can allow non-injective homomorphisms by allowing some ak,l to be zero.) This can be illustrated as a bipartite graph having the vertices marked by numbers (nk)k on one hand and the ones marked by (ml)l on the other hand, and having ak, l edges between the vertex nk and the vertex ml. Thus, when we have a sequence of finite- dimensional semisimple algebras An and injective homomorphisms φn : An' → An+1: between them, we obtain a Bratteli diagram by putting : Vn = the set of simple components of An (each isomorphic to a matrix algebra), marked by the size of matrices.
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Any representation defines a character χ:G → C. Such a function is constant on conjugacy classes of G, a so-called class function; denote the ring of class functions by C(G). If G is finite, the homomorphism R(G) → C(G) is injective, so that R(G) can be identified with a subring of C(G). For fields F whose characteristic divides the order of the group G, the homomorphism from RF(G) → C(G) defined by Brauer characters is no longer injective. For a compact connected group R(G) is isomorphic to the subring of R(T) (where T is a maximal torus) consisting of those class functions that are invariant under the action of the Weyl group (Atiyah and Hirzebruch, 1961).
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Since the set of f for which it can be solved is non-empty, and the set of all f is connected, this shows that it can be solved for all f. The map from smooth functions to smooth functions taking φ to F defined by ::F=(\omega+dd'\phi)^m/\omega^m is neither injective nor surjective. It is not injective because adding a constant to φ does not change F, and it is not surjective because F must be positive and have average value 1. So we consider the map restricted to functions φ that are normalized to have average value 0, and ask if this map is an isomorphism onto the set of positive F=e^f with average value 1.
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It follows that every sheaf E has an injective resolution: :0\to E\to I_0\to I_1\to I_2\to \cdots. Then the sheaf cohomology groups Hi(X,E) are the cohomology groups (the kernel of one homomorphism modulo the image of the previous one) of the complex of abelian groups: : 0\to I_0(X) \to I_1(X) \to I_2(X)\to \cdots. Standard arguments in homological algebra imply that these cohomology groups are independent of the choice of injective resolution of E. The definition is rarely used directly to compute sheaf cohomology. It is nonetheless powerful, because it works in great generality (any sheaf on any topological space), and it easily implies the formal properties of sheaf cohomology, such as the long exact sequence above.
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For real analytic functions, unlike for complex analytic (that is, holomorphic) functions, these statements fail to hold. For example, consider the function :f: (-1, 1) \to (-1, 1) \, given by ƒ(x) = x3. This function is clearly injective, but its derivative is 0 at x = 0, and its inverse is not analytic, or even differentiable, on the whole interval (−1, 1). Consequently, if we enlarge the domain to an open subset G of the complex plane, it must fail to be injective; and this is the case, since (for example) f(εω) = f(ε) (where ω is a primitive cube root of unity and ε is a positive real number smaller than the radius of G as a neighbourhood of 0).
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Injective linear maps between two vector spaces V and W over the same field k induce mappings of the corresponding projective spaces via: ::[v] → [T(v)], where v is a non-zero element of V and [...] denotes the equivalence classes of a vector under the defining identification of the respective projective spaces. Since members of the equivalence class differ by a scalar factor, and linear maps preserve scalar factors, this induced map is well-defined. (If T is not injective, it has a null space larger than {0}; in this case the meaning of the class of T(v) is problematic if v is non-zero and in the null space. In this case one obtains a so-called rational map, see also birational geometry).
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If an operator is not injective (so there is some nonzero x with T(x) = 0), then it is clearly not invertible. So if λ is an eigenvalue of T, one necessarily has λ ∈ σ(T). The set of eigenvalues of T is also called the point spectrum of T, denoted by σp(T).
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If a commutative ring R has prime characteristic p, then we have for all elements x and y in R – the "freshman's dream" holds for power p. The map :f(x) = xp then defines a ring homomorphism :R → R. It is called the Frobenius homomorphism. If R is an integral domain it is injective.
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The invariance of domain theorem states that a continuous and locally injective function between two -dimensional topological manifolds must be open. In functional analysis, the open mapping theorem states that every surjective continuous linear operator between Banach spaces is an open map. This theorem has been generalized to topological vector spaces beyond just Banach spaces.
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Therefore the evaluation homomorphism cannot be injective, and every matrix has a minimal polynomial (not necessarily irreducible). By Cayley–Hamilton theorem, the evaluation homomorphism maps to zero the characteristic polynomial of a matrix. It follows that the minimal polynomial divides the characteristic polynomial, and therefore that the degree of the minimal polynomial is at most .
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The inverse of a function f is often written f^{-1}, but this notation is sometimes ambiguous. Only bijections have two-sided inverses, but any function has a quasi-inverse, i.e., the full transformation monoid is regular. The monoid of partial functions is also regular, whereas the monoid of injective partial transformations is the prototypical inverse semigroup.
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In mathematics, the (right) Ziegler spectrum of a ring R is a topological space whose points are (isomorphism classes of) indecomposable pure-injective right R-modules. Its closed subsets correspond to theories of modules closed under arbitrary products and direct summands. Ziegler spectra are named after Martin Ziegler, who first defined and studied them in 1984.
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This is impossible since is a cardinal such that no injection into exists. Now define a map of into endowed with the lexicographical wellordering by sending to the least such that . By the above reasoning the map exists and is unique since least elements of subsets of wellordered sets are unique. It is, by elementary group theory, injective.
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One distinguishes three separate cases: #T − λ is not injective. That is, there exist two distinct elements x,y in X such that (T − λ)(x) = (T − λ)(y). Then z = x − y is a non-zero vector such that T(z) = λz. In other words, λ is an eigenvalue of T in the sense of linear algebra.
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However, this configuration is not a Garden of Eden because it does have predecessors with infinitely many nonzeros. The fact that every configuration has a predecessor may be summarized by saying that Rule 90 is surjective. The function that maps each configuration to its successor is, mathematically, a surjective function. Rule 90 is also not injective.
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It can be recast to read that a certain natural morphism, the period mapping, from the moduli space of curves of a fixed genus, to a moduli space of abelian varieties, is injective (on geometric points). Generalizations are in two directions. Firstly, to geometric questions about that morphism, for example the local Torelli theorem. Secondly, to other period mappings.
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This can be seen in the following way. A module M embeds into its injective hull E(M), which is now also projective. As a projective module, E(M) is a summand of a free module F, and so E(M) embeds in F with the inclusion map. By composing these two maps, M is embedded in F.
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Although the map w1 : Vect1(X) → H1(X; Z/2Z) is a bijection, the corresponding map is not necessarily injective in higher dimensions. For example, consider the tangent bundle TSn for n even. With the canonical embedding of Sn in Rn+1, the normal bundle ν to Sn is a line bundle. Since Sn is orientable, ν is trivial.
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Asuman Güven Aksoy is a Turkish-American mathematician whose research concerns topics in functional analysis, metric geometry, and operator theory including Banach spaces, measures of non-compactness, fixed points, Birnbaum–Orlicz spaces, real trees, injective metric spaces, and tight spans. She works at Claremont McKenna College, where she is Crown Professor of Mathematics and George R. Roberts Fellow.
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In mathematics, especially in the area of abstract algebra, every module has an associated character module. Using the associated character module it is possible to investigate the properties of the original module. One of the main results discovered by Joachim Lambek shows that a module is flat if and only if the associated character module is injective.
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The Auslander-Reiten quiver of an Artin algebra has a vertex for each indecomposable module and an arrow between vertices if there is an irreducible morphism between the corresponding modules. It has a map τ = D Tr called the translation from the non-projective vertices to the non-injective vertices, where D is the dual and Tr the transpose.
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We give the definition for hypercohomology as this is more common. As usual, hypercohomology and hyperhomology are essentially the same: one converts from one to the other by dualizing, i.e. by changing the direction of all arrows, replacing injective objects with projective ones, and so on. Suppose that A is an abelian category with enough injectives and F a left exact functor to another abelian category B. If C is a complex of objects of A bounded on the left, the hypercohomology :Hi(C) of C (for an integer i) is calculated as follows: # Take a quasi-isomorphism Φ : C → I, here I is a complex of injective elements of A. # The hypercohomology Hi(C) of C is then the cohomology Hi(F(I)) of the complex F(I).
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This means that the cancellative elements of any commutative monoid can be extended to a group. It turns out that requiring the cancellative property in a monoid is not required to perform the Grothendieck construction – commutativity is sufficient. However, if the original monoid has an absorbing element then its Grothendieck group is the trivial group. Hence the homomorphism is, in general, not injective.
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He has over 100 publications listed in the Mathematical Reviews, including 6 books. His earlier work was mostly in module theory, especially torsion theories, non-commutative localization, and injective modules. One of his earliest papers, , proved the Lambek–Moser theorem about integer sequences. In 1963 he published an important result, now known as Lambek's theorem, on character modules characterizing flatness of a module.
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The Schwarz–Ahlfors–Pick theorem provides an analogous theorem for hyperbolic manifolds. De Branges' theorem, formerly known as the Bieberbach Conjecture, is an important extension of the lemma, giving restrictions on the higher derivatives of f at 0 in case f is injective; that is, univalent. The Koebe 1/4 theorem provides a related estimate in the case that f is univalent.
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In mathematics, especially in the field of group theory, a divisible group is an abelian group in which every element can, in some sense, be divided by positive integers, or more accurately, every element is an nth multiple for each positive integer n. Divisible groups are important in understanding the structure of abelian groups, especially because they are the injective abelian groups.
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In mathematics, a metric space aimed at its subspace is a categorical construction that has a direct geometric meaning. It is also a useful step toward the construction of the metric envelope, or tight span, which are basic (injective) objects of the category of metric spaces. Following , a notion of a metric space Y aimed at its subspace X is defined.
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From the standpoint of metric space theory, isometrically isomorphic spaces are identical. ; Isometry: If (M1, d1) and (M2, d2) are metric spaces, an isometry from M1 to M2 is a function f : M1 → M2 such that d2(f(x), f(y)) = d1(x, y) for all x, y in M1. Every isometry is injective, although not every isometry is surjective.
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The shape of the decomposition defines a linear map from coefficient vectors to polynomials of degree less than . The existence proof means that this map is surjective. As the two vector spaces have the same dimension, the map is also injective, which means uniqueness of the decomposition. By the way, this proof induces an algorithm for computing the decomposition through linear algebra.
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A faithful functor need not be injective on objects or morphisms. That is, two objects X and X′ may map to the same object in D (which is why the range of a full and faithful functor is not necessarily isomorphic to C), and two morphisms f : X → Y and f′ : X′ → Y′ (with different domains/codomains) may map to the same morphism in D. Likewise, a full functor need not be surjective on objects or morphisms. There may be objects in D not of the form FX for some X in C. Morphisms between such objects clearly cannot come from morphisms in C. A full and faithful functor is necessarily injective on objects up to isomorphism. That is, if F : C → D is a full and faithful functor and F(X)\cong F(Y) then X \cong Y.
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For example, every function may be factored into the composition of a surjective function with an injective function. Matrices possess many kinds of matrix factorizations. For example, every matrix has a unique LUP factorization as a product of a lower triangular matrix with all diagonal entries equal to one, an upper triangular matrix , and a permutation matrix ; this is a matrix formulation of Gaussian elimination.
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If p\colon X \to Y is a perfect map and if X is locally compact, then Y is locally compact. 4\. If p\colon X \to Y is a perfect map and if X is second countable, then Y is second countable. 5\. Every injective perfect map is a homeomorphism. This follows from the fact that a bijective closed map has a continuous inverse. 6\.
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He introduced, together with Prom Panitchpakdi, the injective metric spaces under the name of "hyperconvex metric spaces". Together with Kennan T. Smith, Aronszajn offered proof of the Aronszajn–Smith theorem. Also, the existence of Aronszajn trees was proven by Aronszajn; Aronszajn lines, also named after him, are the lexicographic orderings of Aronszajn trees. He also made a contribution to the theory of reproducing kernel Hilbert space.
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All operations are byte- oriented. The algorithm uses a single 8×8-bit S-box K, designed so that both K(X) and X XOR K(X) are injective functions. In each round, the bytes of the block are first permuted. Then each byte is XORed with a key byte and an earlier ciphertext byte, processed through the S-box, and XORed with the previous plaintext byte.
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In the other direction, the Garden of Eden theorem for cellular automata implies that every injective update rule is bijective. # The time-reversed dynamics of the automaton can be described by another cellular automaton. Clearly, for this to be possible, the update rule must be bijective. In the other direction, if the update rule is bijective, then it has an inverse function that is also bijective.
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In algebra, a flat cover of a module M over a ring is a surjective homomorphism from a flat module F to M that is in some sense minimal. Any module over a ring has a flat cover that is unique up to (non-unique) isomorphism. Flat covers are in some sense dual to injective hulls, and are related to projective covers and torsion-free covers.
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A plot of the multi-valued imaginary part of the complex logarithm function, which shows the branches. As a complex number z goes around the origin, the imaginary part of the logarithm goes up or down. This makes the origin a branch point of the function. For a function to have an inverse, it must map distinct values to distinct values, that is, it must be injective.
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Monomorphisms are a categorical generalization of injective functions (also called "one-to-one functions"); in some categories the notions coincide, but monomorphisms are more general, as in the examples below. The categorical dual of a monomorphism is an epimorphism, that is, a monomorphism in a category C is an epimorphism in the dual category Cop. Every section is a monomorphism, and every retraction is an epimorphism.
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The words injective, surjective and bijective were introduced to refer to functions which satisfy certain properties.Theory of Sets, p. 84. Bourbaki used simple language for certain geometric objects, naming them pavés (paving stones) and boules (balls) as opposed to "parallelotopes" or "hyperspheroids". Similarly in its treatment of topological vector spaces, Bourbaki defined a barrel as a set which is convex, balanced, absorbing, and closed.
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The performance and success of the overall transmission depends on the parameters of the channel and the block code. Formally, a block code is an injective mapping :C:\Sigma^k \to \Sigma^n. Here, \Sigma is a finite and nonempty set and k and n are integers. The meaning and significance of these three parameters and other parameters related to the code are described below.
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In the above form, the functional to be extended must already be bounded by a sublinear function. In some applications, this might close to begging the question. However, in locally convex spaces, any continuous functional is already bounded by the norm, which is sublinear. One thus hasIn category-theoretic terms, the field is an injective object in the category of locally convex vector spaces.
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This representation is a birational equivalence between the curve and the plane curve defined by f. Every algebraic curve may be represented in this way. However, a linear change of variables may be needed in order to make almost always injective the projection on the two first variables. When a change of variables is needed, almost every change is convenient, as soon as it is defined over an infinite field.
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It is then straightforward to show that contains V and satisfies the above universal property, so that Cl is unique up to a unique isomorphism; thus one speaks of "the" Clifford algebra . It also follows from this construction that i is injective. One usually drops the i and considers V as a linear subspace of . The universal characterization of the Clifford algebra shows that the construction of is functorial in nature.
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Both of these naturalities follow from the naturality of the sequence provided by the snake lemma. Conversely, the following characterization of derived functors holds: given a family of functors Ri: A → B, satisfying the above, i.e. mapping short exact sequences to long exact sequences, such that for every injective object I of A, Ri(I)=0 for every positive i, then these functors are the right derived functors of R0.
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The image of a regular space under an injective, continuous open map is always regular. #Both examples 2 and 3 suggest that Moore spaces are a lot similar to regular spaces. #Neither the Sorgenfrey line nor the Sorgenfrey plane are Moore spaces because they are normal and not second countable. #The Moore plane (also known as the Niemytski space) is an example of a non- metrizable Moore space.
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If M is a supermanifold of dimension (p,q), then the underlying space M inherits the structure of a differentiable manifold whose sheaf of smooth functions is OM/I, where I is the ideal generated by all odd functions. Thus M is called the underlying space, or the body, of M. The quotient map OM → OM/I corresponds to an injective map M → M; thus M is a submanifold of M.
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The algebraic function fields over k form a category; the morphisms from function field K to L are the ring homomorphisms f : K → L with f(a) = a for all a in k. All these morphisms are injective. If K is a function field over k of n variables, and L is a function field in m variables, and n > m, then there are no morphisms from K to L.
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220px In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from to is often denoted with the notation . In the more general setting of category theory, a monomorphism (also called a monic morphism or a mono) is a left-cancellative morphism. That is, an arrow such that for all objects and all morphisms , : f \circ g_1 = f \circ g_2 \implies g_1 = g_2.
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K7, the complete graph with 7 vertices, is a core. Two graphs G and H are homomorphically equivalent if G → H and H → G. The maps are not necessarily surjective nor injective. For instance, the complete bipartite graphs K2,2 and K3,3 are homomorphically equivalent: each map can be defined as taking the left (resp. right) half of the domain graph and mapping to just one vertex in the left (resp.
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A module M over a ring R is called flat if the following condition is satisfied: for any injective map \phi: K \to L of R-modules, the map :K \otimes_R M \to L \otimes_R M induced by k \otimes m \mapsto \phi(k) \otimes m is injective. In other words, for R-modules K, L, and J, if 0\rightarrow K\rightarrow L\rightarrow J\rightarrow 0 is a short exact sequence, then M is a flat module over R if and only if 0\rightarrow K\otimes_R M\rightarrow L\otimes_R M\rightarrow J\otimes_R M\rightarrow 0 is also a short exact sequence. This definition applies also if R is not necessarily commutative, and M is a left R-module and K and L right R-modules. The only difference is that in this case K \otimes_R M and L \otimes_R M are not in general R-modules, but only abelian groups.
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As with metric spaces, every uniform space X has a Hausdorff completion: that is, there exists a complete Hausdorff uniform space Y and a uniformly continuous map i: X → Y with the following property: : for any uniformly continuous mapping f of X into a complete Hausdorff uniform space Z, there is a unique uniformly continuous map g: Y → Z such that f = gi. The Hausdorff completion Y is unique up to isomorphism. As a set, Y can be taken to consist of the minimal Cauchy filters on X. As the neighbourhood filter B(x) of each point x in X is a minimal Cauchy filter, the map i can be defined by mapping x to B(x). The map i thus defined is in general not injective; in fact, the graph of the equivalence relation i(x) = i(x ') is the intersection of all entourages of X, and thus i is injective precisely when X is Hausdorff.
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A fortiori, every module also admits projective and flat resolutions. The proof idea is to define E0 to be the free R-module generated by the elements of M, and then E1 to be the free R-module generated by the elements of the kernel of the natural map E0 -> M etc. Dually, every R-module possesses an injective resolution. Projective resolutions (and, more generally, flat resolutions) can be used to compute Tor functors.
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Universal algebra defines a notion of kernel for homomorphisms between two algebraic structures of the same kind. This concept of kernel measures how far the given homomorphism is from being injective. There is some overlap between this algebraic notion and the categorical notion of kernel since both generalize the situation of groups and modules mentioned above. In general, however, the universal-algebraic notion of kernel is more like the category-theoretic concept of kernel pair.
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In case that mappings arise and arrive on finite sets (discrete bounded value signals), this condition is equivalent to saying that mappings are injective (one-to-one). Moreover, if a mapping goes from one set to a set of the same cardinality, it should be bijective. In the Generalized Lifting Scheme the addition/subtraction restriction is avoided by including this step in the mapping. In this way the Classical Lifting Scheme is generalized.
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Every free abelian group is slender. The additive group of rational numbers Q is not slender: any mapping of the en into Q extends to a homomorphism from the free subgroup generated by the en, and as Q is injective this homomorphism extends over the whole of ZN. Therefore, a slender group must be reduced. Every countable reduced torsion-free abelian group is slender, so every proper subgroup of Q is slender.
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In mathematics, the category Ord has preordered sets as objects and order- preserving functions as morphisms. This is a category because the composition of two order-preserving functions is order preserving and the identity map is order preserving. The monomorphisms in Ord are the injective order-preserving functions. The empty set (considered as a preordered set) is the initial object of Ord, and the terminal objects are precisely the singleton preordered sets.
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In fact, this is necessary and sufficient, because if is any nilpotent, then one of its powers will be nilpotent of order at most . In particular, if is a field then the Frobenius endomorphism is injective. The Frobenius morphism is not necessarily surjective, even when is a field. For example, let be the finite field of elements together with a single transcendental element; equivalently, is the field of rational functions with coefficients in .
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This principle is not a generalization of the pigeonhole principle for finite sets however: It is in general false for finite sets. In technical terms it says that if and are finite sets such that any surjective function from to is not injective, then there exists an element of of such that there exists a bijection between the preimage of and . This is a quite different statement, and is absurd for large finite cardinalities.
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In the case of maps f : U → C defined on an open subset U of the complex plane C, some authors (e.g., Freitag 2009, Definition IV.4.1) define a conformal map to be an injective map with nonzero derivative i.e., f’(z)≠ 0 for every z in U. According to this definition, a map f : U → C is conformal if and only if f: U → f(U) is biholomorphic. Other authors (e.g.
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Eben Matlis (August 28, 1923 - March 27, 2015) was a mathematician known for his contributions to the theory of rings and modules, especially for his work with injective modules over commutative Noetherian rings, and his introduction of Matlis duality. Matlis earned his Ph.D. at the University of Chicago in 1958, with Irving Kaplansky as advisor. He is an emeritus professor at Northwestern University and was a member of the Institute for Advanced Study from August 1962 to June 1963.
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For a separated presheaf, the first arrow need only be injective. Similarly, one can define presheaves and sheaves of abelian groups, rings, modules, and so on. One can require either that a presheaf F is a contravariant functor to the category of abelian groups (or rings, or modules, etc.), or that F be an abelian group (ring, module, etc.) object in the category of all contravariant functors from C to the category of sets. These two definitions are equivalent.
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A set is countable if there exists an injective function from to the natural numbers }.Since there is an obvious bijection between and }, it makes no difference whether one considers 0 a natural number or not. In any case, this article follows ISO 31-11 and the standard convention in mathematical logic, which takes 0 as a natural number. If such an can be found that is also surjective (and therefore bijective), then is called countably infinite.
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Together with Gheorghe Craciun, he developed the theory of injective reaction networks and explored its implications for biochemistry. A current research focus (together with Guy Shinar) is the application of chemical reaction network theory to questions of robustness in biochemical reaction networks. He has also worked with Richard Lavine on foundations of classical thermodynamics. Feinberg is the author of "Foundations of Chemical Reaction Network Theory," published in 2019 by Springer in its Applied Mathematical Sciences series.
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A standard Borel space is the Borel space associated to a Polish space. A standard Borel space is characterized up to isomorphism by its cardinality, and any uncountable standard Borel space has the cardinality of the continuum. For subsets of Polish spaces, Borel sets can be characterized as those sets that are the ranges of continuous injective maps defined on Polish spaces. Note however, that the range of a continuous noninjective map may fail to be Borel.
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A k-tuple point (double, triple, etc.) of an immersion is an unordered set } of distinct points with the same image . If M is an m-dimensional manifold and N is an n-dimensional manifold then for an immersion in general position the set of k-tuple points is an -dimensional manifold. Every embedding is an immersion without multiple points (where ). Note, however, that the converse is false: there are injective immersions that are not embeddings.
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The equivalence relations on any set X, when ordered by set inclusion, form a complete lattice, called Con X by convention. The canonical map ker: X^X → Con X, relates the monoid X^X of all functions on X and Con X. ker is surjective but not injective. Less formally, the equivalence relation ker on X, takes each function f: X→X to its kernel ker f. Likewise, ker(ker) is an equivalence relation on X^X.
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The map :x \bmod N \mapsto (x \bmod n_1, \ldots, x\bmod n_k) maps congruence classes modulo to sequences of congruence classes modulo . The proof of uniqueness shows that this map is injective. As the domain and the codomain of this map have the same number of elements, the map is also surjective, which proves the existence of the solution. This proof is very simple but does not provide any direct way for computing a solution.
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Let K be an algebraic number field, and let OK be its ring of integers. Let b1, ..., bn be an integral basis of OK (i.e. a basis as a Z-module), and let {σ1, ..., σn} be the set of embeddings of K into the complex numbers (i.e. injective ring homomorphisms K → C). The discriminant of K is the square of the determinant of the n by n matrix B whose (i,j)-entry is σi(bj).
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A correspondence between transformation semigroups and semigroup actions is described below. If we restrict it to faithful semigroup actions, it has nice properties. Any transformation semigroup can be turned into a semigroup action by the following construction. For any transformation semigroup S of X, define a semigroup action T of S on X as T(s, x) = s(x) for s\in S, x\in X. This action is faithful, which is equivalent to curry(T) being injective.
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Formally, if is a set, the identity function on is defined to be that function with domain and codomain which satisfies : for all elements in . In other words, the function value in (that is, the codomain) is always the same input element of (now considered as the domain). The identity function on is clearly an injective function as well as a surjective function, so it is also bijective. The identity function on is often denoted by .
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A bijective function, f: X → Y, where set X is {1, 2, 3, 4} and set Y is {A, B, C, D}. For example, f(1) = D. In mathematics, a bijection, bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set. There are no unpaired elements. In mathematical terms, a bijective function is a one-to-one (injective) and onto (surjective) mapping of a set X to a set Y. The term one- to-one correspondence must not be confused with one-to-one function (an injective function; see figures). A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements.
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In mathematics, in the field of complex analysis, a Nevanlinna function is a complex function which is an analytic function on the open upper half-plane H and has non-negative imaginary part. A Nevanlinna function maps the upper half-plane to itself or to a real constant,A real number is not considered to be in the upper half-plane. but is not necessarily injective or surjective. Functions with this property are sometimes also known as Herglotz, Pick or R functions.
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But the complex exponential function is not injective, because for any w, since adding iθ to w has the effect of rotating ew counterclockwise θ radians. So the points :\ldots,\;w-4\pi i, \;w-2\pi i, \;w, \;w + 2\pi i, \;w+4\pi i, \;\ldots, equally spaced along a vertical line, are all mapped to the same number by the exponential function. This means that the exponential function does not have an inverse function in the standard sense.Conway, p. 39.
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If the original group is linear, so too is the universal complexification and the homomorphism between the two is an inclusion. give an example of a connected real Lie group for which the homomorphism is not injective even at the Lie algebra level: they take the product of by the universal covering group of and quotient out by the discrete cyclic subgroup generated by an irrational rotation in the first factor and a generator of the center in the second.
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Suppose that R is a Noetherian complete local ring with residue field k, and choose E to be an injective hull of k (sometimes called a Matlis module). The dual DR(M) of a module M is defined to be HomR(M,E). Then Matlis duality states that the duality functor DR gives an anti-equivalence between the categories of Artinian and Noetherian R-modules. In particular the duality functor gives an anti-equivalence from the category of finite-length modules to itself.
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Unlike a regular manifold, a supermanifold is not entirely composed of a set of points. Instead, one takes the dual point of view that the structure of a supermanifold M is contained in its sheaf OM of "smooth functions". In the dual point of view, an injective map corresponds to a surjection of sheaves, and a surjective map corresponds to an injection of sheaves. An alternative approach to the dual point of view is to use the functor of points.
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UV coordinates are optionally applied per face. This means a shared spatial vertex position can have different UV coordinates for each of its triangles, so adjacent triangles can be cut apart and positioned on different areas of the texture map. The UV mapping process at its simplest requires three steps: unwrapping the mesh, creating the texture, and applying the texture to a respective face of polygon. UV mapping may use repeating textures, or an injective 'unique' mapping as a prerequisite for baking.
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The case where ordering doesn't matter, however, is comparable to describing a single multinomial distribution of N draws from an X-fold category, where only the number seen of each category matters. The case where ordering doesn't matter and sampling is without replacement is comparable to a single multivariate hypergeometric distribution, and the fourth possibility does not seem to have a correspondence. Note that in all the "injective" cases (i.e. sampling without replacement), the number of sets of choices is zero unless .
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In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero such that , or equivalently if the map from to that sends to is not injective (one to one). Similarly, an element of a ring is called a right zero divisor if there exists a nonzero such that . This is a partial case of divisibility in rings. An element that is a left or a right zero divisor is simply called a zero divisor.
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In mathematics, an embedding (or imbedding suggests that "the English" (i.e. the British) use "embedding" instead of "imbedding".) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup. When some object X is said to be embedded in another object Y, the embedding is given by some injective and structure-preserving map . The precise meaning of "structure-preserving" depends on the kind of mathematical structure of which X and Y are instances.
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See for example Fig. 4; there, the arrow from "real linear topological" to "real linear" is two-headed, since every real linear space admits some (at least one) topology compatible with its linear structure. Such topology is non-unique in general, but unique when the real linear space is finite- dimensional. For these spaces the transition is both injective and surjective, that is, bijective; see the arrow from "finite-dim real linear topological" to "finite-dim real linear" on Fig. 4.
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By the lifting property the map f lifts to a continuous map such that the restriction of g to the boundary S1 of D2 is equal to γ. Therefore, γ is null-homotopic in C, so that the kernel of is trivial and thus is an injective homomorphism. Therefore, is isomorphic to the subgroup of . If is another pre-image of x in C then the subgroups and are conjugate in by p-image of a curve in C connecting c to c1.
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In commutative algebra, a Gorenstein local ring is a commutative Noetherian local ring R with finite injective dimension as an R-module. There are many equivalent conditions, some of them listed below, often saying that a Gorenstein ring is self-dual in some sense. Gorenstein rings were introduced by Grothendieck in his 1961 seminar (published in ). The name comes from a duality property of singular plane curves studied by (who was fond of claiming that he did not understand the definition of a Gorenstein ring).
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Invariance of domain is a theorem in topology about homeomorphic subsets of Euclidean space . It states: :If is an open subset of and is an injective continuous map, then is open in and is a homeomorphism between and . The theorem and its proof are due to L. E. J. Brouwer, published in 1912. Beweis der Invarianz des n-dimensionalen Gebiets, Mathematische Annalen 71 (1912), pages 305–315; see also 72 (1912), pages 55–56 The proof uses tools of algebraic topology, notably the Brouwer fixed point theorem.
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In mathematics, the Honda–Tate theorem classifies abelian varieties over finite fields up to isogeny. It states that the isogeny classes of simple abelian varieties over a finite field of order q correspond to algebraic integers all of whose conjugates (given by eigenvalues of the Frobenius endomorphism on the first cohomology group or Tate module) have absolute value . showed that the map taking an isogeny class to the eigenvalues of the Frobenius is injective, and showed that this map is surjective, and therefore a bijection.
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Kernels allow defining quotient objects (also called quotient algebras in universal algebra, and cokernels in category theory). For many types of algebraic structure, the fundamental theorem on homomorphisms (or first isomorphism theorem) states that image of a homomorphism is isomorphic to the quotient by the kernel. The concept of a kernel has been extended to structures such that the inverse image of a single element is not sufficient for deciding whether an homomorphism is injective. In these cases, the kernel is a congruence relation.
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PWPP is the corresponding class of problems that are polynomial-time reducible to it. WEAK-PIGEON is the following problem: :Given a Boolean circuit C having n input bits and n-1 output bits, find x e y such that C(x) = C(y). Here, the range of the circuit is strictly smaller than its domain, so the circuit is guaranteed to be non-injective. WEAK-PIGEON reduces to PIGEON by appending a single 1 bit to the circuit's output, so PWPP \subseteq PPP.
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In both cases the third class of morphisms is given by a lifting condition (see below). In some cases, when the category C is a Reedy category, there is a third model structure lying in between the projective and injective. The process of forcing certain maps to become weak equivalences in a new model category structure on the same underlying category is known as Bousfield localization. For example, the category of simplicial sheaves can be obtained as a Bousfield localization of the model category of simplicial presheaves.
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The map is either an isomorphism (the image is the whole group), or an injective map with index 2. The latter is the case if and only if there exists an n-dimensional framed manifold with Kervaire invariant 1, which is known as the Kervaire invariant problem. Thus a factor of 2 in the classification of exotic spheres depends on the Kervaire invariant problem. , the Kervaire invariant problem is almost completely solved, with only the case n=126 remaining open; see that article for details.
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There are currently no successful methods in controlling the severity and occurrence of the disease on a commercial scale ,. No fungicides and soil fumigation strategies have proved successful. Chemicals such as methyl bromide, anhydrous ammonia and ammonium salts were tested as effective but required deep injective into the soil, therefore expensive and not commercially feasible. In addition, systemic fungicides such as benzimidazoles and sterol biosynthesis inhibitors were shown to limit P. omnivore as well, but systemic fungicides are expensive, and exhibit poor soil penetration therefore pursued cautiously.
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On any topological space, the skyscraper sheaf associated to a closed point x and a group or ring G has the stalks 0 off x and G in x--hence the name skyscraper. The same property holds for any point x if the topological space in question is a T1 space, since every point of a T1 space is closed. This feature is the basis of the construction of Godement resolutions, used for example in algebraic geometry to get functorial injective resolutions of sheaves.
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Suppose that k is a perfect field, with ring of Witt vectors W and let K be the quotient field of W, with Frobenius automorphism σ. Over the field k, an F-crystal is a free module M of finite rank over the ring W of Witt vectors of k, together with a σ-linear injective endomorphism of M. An F-isocrystal is defined in the same way, except that M is a module for the quotient field K of W rather than W.
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The notion of one-to-one correspondence generalizes to partial functions, where they are called partial bijections, although partial bijections are only required to be injective. The reason for this relaxation is that a (proper) partial function is already undefined for a portion of its domain; thus there is no compelling reason to constrain its inverse to be a total function, i.e. defined everywhere on its domain. The set of all partial bijections on a given base set is called the symmetric inverse semigroup.
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If is a homomorphism of rings of characteristic , then :\phi(x^p) = \phi(x)^p. If and are the Frobenius endomorphisms of and , then this can be rewritten as: :\phi \circ F_R = F_S \circ \phi. This means that the Frobenius endomorphism is a natural transformation from the identity functor on the category of characteristic rings to itself. If the ring is a ring with no nilpotent elements, then the Frobenius endomorphism is injective: means , which by definition means that is nilpotent of order at most .
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The strong reducibilities include: ;One-one reducibility: A is one-one reducible (or 1-reducible) to B if there is a total computable injective function f such that each n is in A if and only if f(n) is in B. ;Many-one reducibility: This is essentially one-one reducibility without the constraint that f be injective. A is many-one reducible (or m-reducible) to B if there is a total computable function f such that each n is in A if and only if f(n) is in B. ;Truth-table reducibility: A is truth-table reducible to B if A is Turing reducible to B via an oracle Turing machine that computes a total function regardless of the oracle it is given. Because of compactness of Cantor space, this is equivalent to saying that the reduction presents a single list of questions (depending only on the input) to the oracle simultaneously, and then having seen their answers is able to produce an output without asking additional questions regardless of the oracle's answer to the initial queries. Many variants of truth-table reducibility have also been studied.
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In computer science, a charging argument is used to compare the output of an optimization algorithm to an optimal solution. It is typically used to show that an algorithm produces optimal results by proving the existence of a particular injective function. For profit maximization problems, the function can be any one-to-one mapping from elements of an optimal solution to elements of the algorithm's output. For cost minimization problems, the function can be any one-to-one mapping from elements of the algorithm's output to elements of an optimal solution.
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In mathematics, the category of magmas, denoted Mag, is the category whose objects are magmas (that is, sets equipped with a binary operation), and whose morphisms are magma homomorphisms. The category Mag has direct products, so the concept of a magma object (internal binary operation) makes sense. (As in any category with direct products.) There is an inclusion functor: as trivial magmas, with operations given by projection: . An important property is that an injective endomorphism can be extended to an automorphism of a magma extension, just the colimit of the (constant sequence of the) endomorphism.
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An (E, G)-ring B is called regular if #B is reduced; #for every V in Rep(G), αB,V is injective; #every b ∈ B for which the line bE is G-stable is invertible in B. The third condition implies F is a field. If B is a field, it is automatically regular. When B is regular, :\dim_FD_B(V)\leq\dim_EV with equality if, and only if, αB,V is an isomorphism. A representation V ∈ Rep(G) is called B-admissible if αB,V is an isomorphism.
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In metric geometry, an injective metric space, or equivalently a hyperconvex metric space, is a metric space with certain properties generalizing those of the real line and of L∞ distances in higher-dimensional vector spaces. These properties can be defined in two seemingly different ways: hyperconvexity involves the intersection properties of closed balls in the space, while injectivity involves the isometric embeddings of the space into larger spaces. However it is a theorem of Aronszajn and Panitchpakdi (1956; see e.g. ) that these two different types of definitions are equivalent.
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The domain invariance theorem may be generalized to manifolds: if and are topological -manifolds without boundary and is a continuous map which is locally one-to-one (meaning that every point in has a neighborhood such that restricted to this neighborhood is injective), then is an open map (meaning that is open in whenever is an open subset of ) and a local homeomorphism. There are also generalizations to certain types of continuous maps from a Banach space to itself. Topologie des espaces abstraits de M. Banach. C. R. Acad. Sci.
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Generally, the minimum number of parameters required to describe a model or geometric object is equal to its dimension, and the scope of the parameters—within their allowed ranges—is the parameter space. Though a good set of parameters permits identification of every point in the object space, it may be that, for a given parametrization, different parameter values can refer to the same point. Such mappings are surjective but not injective. An example is the pair of cylindrical polar coordinates (ρ, φ, z) and (ρ, φ + 2π, z).
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The forgetful functor has a left adjoint (which associates to a given set the free abelian group with that set as basis) but does not have a right adjoint. Taking direct limits in Ab is an exact functor. Since the group of integers Z serves as a generator, the category Ab is therefore a Grothendieck category; indeed it is the prototypical example of a Grothendieck category. An object in Ab is injective if and only if it is a divisible group; it is projective if and only if it is a free abelian group.
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Definition 2.1. of . Given any category C and a model category M, under certain extra hypothesis the category of functors Fun (C, M) (also called C-diagrams in M) is also a model category. In fact, there are always two candidates for distinct model structures: in one, the so-called projective model structure, fibrations and weak equivalences are those maps of functors which are fibrations and weak equivalences when evaluated at each object of C. Dually, the injective model structure is similar with cofibrations and weak equivalences instead.
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The algebraic approach to graph rewriting is based upon category theory. The algebraic approach is further divided into sub-approaches, the most common of which are the double-pushout (DPO) approach and the single-pushout (SPO) approach. Other sub-approaches include the sesqui-pushout and the pullback approach. From the perspective of the DPO approach a graph rewriting rule is a pair of morphisms in the category of graphs and graph homomorphisms between them: r = (L \leftarrow K \rightarrow R) (or L \supseteq K \subseteq R) where K \rightarrow L is injective.
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Given a finitely generated module over a commutative ring, the pairing may be injective (hence "nondegenerate" in the above sense) but not unimodular. For example, over the integers, the pairing is nondegenerate but not unimodular, as the induced map from to is multiplication by 2. If V is finite-dimensional then one can identify V with its double dual V∗∗. One can then show that B2 is the transpose of the linear map B1 (if V is infinite-dimensional then B2 is the transpose of B1 restricted to the image of V in V∗∗).
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The `TypeFamilies` extension in the Glasgow Haskell Compiler supports both type synonym families and data families. Type synonym families are the more flexible (but harder to type- check) form, permitting the types in the codomain of the type function to be any type whatsoever with the appropriate kind. Data families, on the other hand, restrict the codomain by requiring each instance to define a new type constructor for the function's result. This ensures that the function is injective, allowing clients' contexts to deconstruct the type family and obtain the original argument type.
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In the latter case, however, once we've chosen an item, we put it aside so that we can't choose it again. This means that the act of choosing an item has an effect on all the following choices (the particular item can't be seen again), so our choices are dependent on one another. In the terminology below, the case of sampling with replacement is termed "Any f", while the case of sampling without replacement is termed "Injective f". Each box indicates how many different sets of choices there are, in a particular sampling scheme.
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Three iterations of a Peano curve construction, whose limit is a space-filling curve. In geometry, the Peano curve is the first example of a space-filling curve to be discovered, by Giuseppe Peano in 1890.. Peano's curve is a surjective, continuous function from the unit interval onto the unit square, however it is not injective. Peano was motivated by an earlier result of Georg Cantor that these two sets have the same cardinality. Because of this example, some authors use the phrase "Peano curve" to refer more generally to any space-filling curve..
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In linear and homological algebra, a monad is a 3-term complex : A → B → C of objects in some abelian category whose middle term B is projective and whose first map A → B is injective and whose second map B → C is surjective. Equivalently a monad is a projective object together with a 3-step filtration (B ⊃ ker(B → C) ⊃ im(A → B)). In practice A, B, and C are often vector bundles over some space, and there are several minor extra conditions that some authors add to the definition. Monads were introduced by .
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If a family of nonempty sets has an empty intersection, its Helly number must be at least two, so the smallest k for which the k-Helly property is nontrivial is k = 2. The 2-Helly property is also known as the Helly property. A 2-Helly family is also known as a Helly family. A convex metric space in which the closed balls have the 2-Helly property (that is, a space with Helly dimension 1, in the stronger variant of Helly dimension for infinite subcollections) is called injective or hyperconvex.
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When the above partial order is restricted to the central idempotents of R, a lattice structure, or even a Boolean algebra structure, can be given. For two central idempotents e and f the complement and the join and meet are given by :e ∨ f = e + f − ef and :e ∧ f = ef. The ordering now becomes simply if and only if , and the join and meet satisfy and . It is shown in that if R is von Neumann regular and right self- injective, then the lattice is a complete lattice.
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In mathematics, the Ax–Grothendieck theorem is a result about injectivity and surjectivity of polynomials that was proved independently by James Ax and Alexander Grothendieck... The theorem is often given as this special case: If P is an injective polynomial function from an n-dimensional complex vector space to itself then P is bijective. That is, if P always maps distinct arguments to distinct values, then the values of P cover all of Cn. The full theorem generalizes to any algebraic variety over an algebraically closed field.Éléments de géométrie algébrique, IV3, Proposition 10.4.11.
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Let E ⊂ g(K) be non-measurable, and let F = g−1(E). Because g is injective, we have that F ⊂ K, and so F is a null set. However, if it were Borel measurable, then g(F) would also be Borel measurable (here we use the fact that the preimage of a Borel set by a continuous function is measurable; g(F) = (g−1)−1(F) is the preimage of F through the continuous function h = g−1.) Therefore, F is a null, but non-Borel measurable set.
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If X is a Tychonoff space then the map from X to its image in βX is a homeomorphism, so X can be thought of as a (dense) subspace of βX; every other compact Hausdorff space that densely contains X is a quotient of βX. For general topological spaces X, the map from X to βX need not be injective. A form of the axiom of choice is required to prove that every topological space has a Stone–Čech compactification. Even for quite simple spaces X, an accessible concrete description of βX often remains elusive.
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Morton and Hilbert curves of level 6 (45=1024 cells in the recursive square partition) plotting each address as different color in the RGB standard, and using Geohash labels. The neighborhoods have similar colors, but each curve offers different pattern of grouping similars in smaller scales. If a curve is not injective, then one can find two intersecting subcurves of the curve, each obtained by considering the images of two disjoint segments from the curve's domain (the unit line segment). The two subcurves intersect if the intersection of the two images is non-empty.
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The situation is the same with preimages under non-injective maps (even if the map is surjective). If is a proper subset then any filter on will not be a filter on , although it will be a prefilter. One advantage that filters have is that they are distinguished representatives of their equivalence class (relative to ), meaning that any equivalence class of prefilters contains a unique filter. This property may be useful when dealing with equivalence classes of prefilters (for instance, they're useful in construct completions using Cauchy filters).
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If R is a Gorenstein ring, then R considered as a module over itself is a dualizing module. If R is an Artinian local ring then the Matlis module of R (the injective hull of the residue field) is the dualizing module. The Artinian local ring R = k[x,y]/(x2,y2,xy) has a unique dualizing module, but it is not isomorphic to R. The ring Z[] has two non-isomorphic dualizing modules, corresponding to the two classes of invertible ideals. The local ring k[x,y]/(y2,xy) is not Cohen–Macaulay so does not have a dualizing module.
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The Hasse principle for algebraic groups states that if G is a simply-connected algebraic group defined over the global field k then the map from : H^1(k,G)\rightarrow\prod_s H^1(k_s,G) is injective, where the product is over all places s of k. The Hasse principle for orthogonal groups is closely related to the Hasse principle for the corresponding quadratic forms. and several others verified the Hasse principle by case-by-case proofs for each group. The last case was the group E8 which was only completed by many years after the other cases.
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Openness of V in the subspace topology is automatic.) Both of these statements are not at all obvious and are not generally true if one leaves Euclidean space. A map which is not a homeomorphism onto its image: with g(t) = (t2 − 1, t3 − t) It is of crucial importance that both domain and range of f are contained in Euclidean space of the same dimension. Consider for instance the map f : (0,1) → ℝ2 defined by . This map is injective and continuous, the domain is an open subset of , but the image is not open in .
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Because some distinct pairs of antipodes are all taken to identical points in the Roman surface, it is not homeomorphic to RP2, but is instead a quotient of the real projective plane RP2 = S2 / (x~-x). Furthermore, the map T (above) from S2 to this quotient has the special property that it is locally injective away from six pairs of antipodal points. Or from RP2 the resulting map making this an immersion of RP2 — minus six points — into 3-space. (It was previously stated that the Roman surface is homeomorphic to RP2, but this was in error.
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In algebra, the kernel of a homomorphism (function that preserves the structure) is generally the inverse image of 0 (except for groups whose operation is denoted multiplicatively, where the kernel is the inverse image of 1). An important special case is the kernel of a linear map. The kernel of a matrix, also called the null space, is the kernel of the linear map defined by the matrix. The kernel of a homomorphism is reduced to 0 (or 1) if and only if the homomorphism is injective, that is if the inverse image of every element consists of a single element.
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Conversely, every injection f with non-empty domain has a left inverse g, which can be defined by fixing an element a in the domain of f so that g(x) equals the unique preimage of x under f if it exists and g(x) = a otherwise. The left inverse g is not necessarily an inverse of f, because the composition in the other order, , may differ from the identity on Y. In other words, an injective function can be "reversed" by a left inverse, but is not necessarily invertible, which requires that the function is bijective.
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A perfect hash function for the four names shown A minimal perfect hash function for the four names shown In computer science, a perfect hash function for a set is a hash function that maps distinct elements in to a set of integers, with no collisions. In mathematical terms, it is an injective function. Perfect hash functions may be used to implement a lookup table with constant worst-case access time. A perfect hash function has many of the same applications as other hash functions, but with the advantage that no collision resolution has to be implemented.
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A variety of general type X is one of maximal Kodaira dimension (Kodaira dimension equal to its dimension): :\kappa(X) = \dim\ X. Equivalent conditions are that the line bundle K_X is big, or that the d-canonical map is generically injective (that is, a birational map to its image) for d sufficiently large. For example, a variety with ample canonical bundle is of general type. In some sense, most algebraic varieties are of general type. For example, a smooth hypersurface of degree d in the n-dimensional projective space is of general type if and only if d > n+1.
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The bag can be wrapped more than once by twisting it and wrapping it back over the ball. (There is no requirement for the continuous map to be injective and so the bag is allowed to pass through itself.) The twist can be in one of two directions and opposite twists can cancel out by deformation. The total number of twists after cancellation is an integer, called the degree of the mapping. As in the case mappings from the circle to the circle, this degree identifies the homotopy group with the group of integers, ℤ. These two results generalize: for all , (see below).
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A Seifert surface S for an oriented link L is an oriented surface whose boundary is L with the same induced orientation. If S is not π1 injective in S3 − N(L), where N(L) is a tubular neighborhood of L, then the loop theorem gives a compressing disk that one may use to compress S along, providing another Seifert surface of reduced complexity. Hence, there are incompressible Seifert surfaces. Every Seifert surface of a link is related to one another through compressions in the sense that the equivalence relation generated by compression has one equivalence class.
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If G is a finite group and k a field with characteristic 0, then one shows in the theory of group representations that any subrepresentation of a given one is already a direct summand of the given one. Translated into module language, this means that all modules over the group algebra kG are injective. If the characteristic of k is not zero, the following example may help. If A is a unital associative algebra over the field k with finite dimension over k, then Homk(−, k) is a duality between finitely generated left A-modules and finitely generated right A-modules.
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A curve \gamma is closedThis term my be ambiguous, as a non-closed curve may be a closed set, as is a line in a plane or is a loop if I = [a, b] and \gamma(a) = \gamma(b). A closed curve is thus the image of a continuous mapping of a circle. If the domain of \gamma is a closed and bounded interval I = [a, b], the curve is also called a path or an arc. A curve is simple if it is the image of an interval or a circle by an injective continuous function.
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In this case, λ is said to be in the point spectrum of T, denoted σp(T). #T − λ is injective, and its range is a dense subset R of X; but is not the whole of X. In other words, there exists some element x in X such that (T − λ)(y) can be as close to x as desired, with y in X; but is never equal to x. It can be proved that, in this case, T − λ is not bounded below (i.e. it sends far apart elements of X too close together).
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When viewing ƒ as a labelling of the elements of N, the latter may be thought of as arranged in a sequence, and the labels as being successively assigned to them. A requirement that ƒ be injective means that no label can be used a second time; the result is a sequence of labels without repetition. In the absence of such a requirement, the terminology "sequences with repetition" is used, meaning that labels may be used more than once (although sequences that happen to be without repetition are also allowed). For an unordered selection the same kind of distinction applies.
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When viewing ƒ as a grouping of the elements of N (which assumes one identifies under permutations of X), requiring ƒ to be surjective means the number of groups must be exactly x. Without this requirement the number of groups can be at most x. The requirement of injective ƒ means each element of N must be a group in itself, which leaves at most one valid grouping and therefore gives a rather uninteresting counting problem. When in addition one identifies under permutations of N, this amounts to forgetting the groups themselves but retaining only their sizes.
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Then Ext(A, B) is the cohomology of this complex at position i. Cartan and Eilenberg showed that these constructions are independent of the choice of projective or injective resolution, and that both constructions yield the same Ext groups.Weibel (1994), sections 2.4 and 2.5 and Theorem 2.7.6. Moreover, for a fixed ring R, Ext is a functor in each variable (contravariant in A, covariant in B). For a commutative ring R and R-modules A and B, Ext(A, B) is an R-module (using that HomR(A, B) is an R-module in this case).
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Music memory games test a player's musical memory. Sight-reading music games take a variety of forms depending upon which aspect of the music serves as the focus of gameplay. Although the majority of such games primarily emphasize rhythm as the major gameplay-determinative musical element, other elements of musical notation and development such as pitch and volume also serve as points of emphasis in a number of games. In all of these game-forms the goal of the player is to provide a direct injective response to each prompt (linked to an element of the music) from the game.
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First, when V is not locally convex, the continuous dual may be equal to {0} and the map Ψ trivial. However, if V is Hausdorff and locally convex, the map Ψ is injective from V to the algebraic dual of the continuous dual, again as a consequence of the Hahn–Banach theorem.If V is locally convex but not Hausdorff, the kernel of Ψ is the smallest closed subspace containing {0}. Second, even in the locally convex setting, several natural vector space topologies can be defined on the continuous dual , so that the continuous double dual is not uniquely defined as a set.
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Let Γ be a finite connected graph. A combinatorial map f : Γ → Γ is called a train track map if for every edge e of Γ and every integer n ≥ 1 the edge-path fn(e) contains no backtracks, that is, it contains no subpaths of the form hh−1 where h is an edge of Γ. In other words, the restriction of fn to e is locally injective (or an immersion) for every edge e and every n ≥ 1\. When applied to the case n = 1, this definition implies, in particular, that the path f(e) has no backtracks.
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Grothendieck's proof of the theorem is based on proving the analogous theorem for finite fields and their algebraic closures. That is, for any field F that is itself finite or that is the closure of a finite field, if a polynomial P from Fn to itself is injective then it is bijective. If F is a finite field, then Fn is finite. In this case the theorem is true for trivial reasons having nothing to do with the representation of the function as a polynomial: any injection of a finite set to itself is a bijection.
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Then the groups Hi(X,E) for integers i are defined as the right derived functors of the functor E ↦ E(X). This makes it automatic that Hi(X,E) is zero for i < 0, and that H0(X,E) is the group E(X) of global sections. The long exact sequence above is also straightforward from this definition. The definition of derived functors uses that the category of sheaves of abelian groups on any topological space X has enough injectives; that is, for every sheaf E there is an injective sheaf I with an injection E → I.Iversen (1986), Theorem II.3.1.
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Suppose V is a non-trivial variety of algebras, i.e. V contains algebras with more than one element. One can show that for every set S, the variety V contains a free algebra FS on S. This means that there is an injective set map i : S -> FS which satisfies the following universal property: given any algebra A in V and any map k : S -> A, there exists a unique V-homomorphism f : FS -> A such that f\circ i = k. This generalizes the notions of free group, free abelian group, free algebra, free module etc.
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Diagram chasing (also called diagrammatic search) is a method of mathematical proof used especially in homological algebra, where one establishes a property of some morphism by tracing the elements of a commutative diagram. A proof by diagram chasing typically involves the formal use of the properties of the diagram, such as injective or surjective maps, or exact sequences. A syllogism is constructed, for which the graphical display of the diagram is just a visual aid. It follows that one ends up "chasing" elements around the diagram, until the desired element or result is constructed or verified.
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In the language of category theory, any universal construction gives rise to a functor; one thus obtains a functor from the category of commutative monoids to the category of abelian groups which sends the commutative monoid M to its Grothendieck group K. This functor is left adjoint to the forgetful functor from the category of abelian groups to the category of commutative monoids. For a commutative monoid M, the map i : M->K is injective if and only if M has the cancellation property, and it is bijective if and only if M is already a group.
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The space of uniform almost periodic functions on G can be identified with the space of all continuous functions on the Bohr compactification of G. More generally the Bohr compactification can be defined for any topological group G, and the spaces of continuous or Lp functions on the Bohr compactification can be considered as almost periodic functions on G. For locally compact connected groups G the map from G to its Bohr compactification is injective if and only if G is a central extension of a compact group, or equivalently the product of a compact group and a finite-dimensional vector space.
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The topologies of locally convex topological vector spaces A and B are given by families of seminorms. For each choice of seminorm on A and on B we can define the corresponding family of cross norms on the algebraic tensor product A\otimes B, and by choosing one cross norm from each family we get some cross norms on A\otimes B, defining a topology. There are in general an enormous number of ways to do this. The two most important ways are to take all the projective cross norms, or all the injective cross norms.
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In particular, if the space Z is assumed to be simply connected (so that is trivial), condition is automatically satisfied, and every continuous map from Z to X can be lifted. Since the unit interval is simply connected, the lifting property for paths is a special case of the lifting property for maps stated above. If is a covering and and are such that , then p# is injective at the level of fundamental groups, and the induced homomorphisms are isomorphisms for all . Both of these statements can be deduced from the lifting property for continuous maps.
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Let be a covering map where both X and C are path-connected. Let be a basepoint of X and let be one of its pre-images in C, that is . There is an induced homomorphism of fundamental groups which is injective by the lifting property of coverings. Specifically if γ is a closed loop at c such that , that is is null-homotopic in X, then consider a null-homotopy of as a map from the 2-disc D2 to X such that the restriction of f to the boundary S1 of D2 is equal to .
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A retraction of a metric space X is a function ƒ mapping X to a subspace of itself, such that # for all x, ƒ(ƒ(x)) = ƒ(x); that is, ƒ is the identity function on its image (i. e. it is idempotent), and # for all x and y, d(ƒ(x), ƒ(y)) ≤ d(x, y); that is, ƒ is nonexpansive. A retract of a space X is a subspace of X that is an image of a retraction. A metric space X is said to be injective if, whenever X is isometric to a subspace Z of a space Y, that subspace Z is a retract of Y.
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In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements of its codomain. In other words, every element of the function's codomain is the image of at most one element of its domain. The term one-to- one function must not be confused with one-to-one correspondence that refers to bijective functions, which are functions such that each element in the codomain is an image of exactly one element in the domain. A homomorphism between algebraic structures is a function that is compatible with the operations of the structures.
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The objects are the topological vector spaces over and the morphisms are the continuous -linear maps from one object to another. :Definition: A TVS homomorphism or topological homomorphism is a continuous linear map between topological vector spaces (TVSs) such that the induced map is an open mapping when , which is the range or image of , is given the subspace topology induced by Y. :Definition: A TVS embedding or a topological monomorphism is an injective topological homomorphism. Equivalently, a TVS-embedding is a linear map that is also a topological embedding. :Definition: A TVS isomorphism or an isomorphism in the category of TVSs is a bijective linear homeomorphism.
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In computer science, a type family associates data types with other data types, using a type-level function defined by an open-ended collection of valid instances of input types and the corresponding output types. Type families are a feature of some type systems that allow partial functions between types to be defined by pattern matching. This is in contrast to data type constructors, which define injective functions from all types of a particular kind to a new set of types, and type synonyms (a.k.a. typedef), which define functions from all types of a particular kind to another existing set of types using a single case.
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The vector space V is called the representation space of φ and its dimension (if finite) is called the dimension of the representation (sometimes degree, as in .). It is also common practice to refer to V itself as the representation when the homomorphism φ is clear from the context; otherwise the notation (V,φ) can be used to denote a representation. When V is of finite dimension n, one can choose a basis for V to identify V with Fn, and hence recover a matrix representation with entries in the field F. An effective or faithful representation is a representation (V,φ), for which the homomorphism φ is injective.
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In many categories it is possible to write every morphism as the composition of an epimorphism followed by a monomorphism. For instance, given a group homomorphism f : G → H, we can define the group K = im(f) and then write f as the composition of the surjective homomorphism G → K that is defined like f, followed by the injective homomorphism K → H that sends each element to itself. Such a factorization of an arbitrary morphism into an epimorphism followed by a monomorphism can be carried out in all abelian categories and also in all the concrete categories mentioned above in (though not in all concrete categories).
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On the other hand, the set of first-order sentences valid in the reals has arbitrarily large models due to the compactness theorem. Thus the least-upper-bound property cannot be expressed by any set of sentences in first-order logic. (In fact, every real- closed field satisfies the same first-order sentences in the signature \langle +,\cdot,\le\rangle as the real numbers.) In second-order logic, it is possible to write formal sentences which say "the domain is finite" or "the domain is of countable cardinality." To say that the domain is finite, use the sentence that says that every surjective function from the domain to itself is injective.
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Note that in an infinite dimensional space, we can have a bilinear form ƒ for which v \mapsto (x \mapsto f(x,v)) is injective but not surjective. For example, on the space of continuous functions on a closed bounded interval, the form : f(\phi,\psi) = \int\psi(x)\phi(x) dx is not surjective: for instance, the Dirac delta functional is in the dual space but not of the required form. On the other hand, this bilinear form satisfies :f(\phi,\psi)=0\, for all \,\phi implies that \psi=0.\, In such a case where ƒ satisfies injectivity (but not necessarily surjectivity), ƒ is said to be weakly nondegenerate.
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Alternatively, every join-prime closed set is the closure of a unique point, where "join-prime" can be replaced by (join-) irreducible since we are in a distributive lattice. Spaces with this property are called sober. Conversely, for a locale L, φ: L → Ω(pt(L)) is always surjective. It is additionally injective if and only if any two elements a and b of L for which a is not less-or-equal to b can be separated by points of the locale, formally: : if not a ≤ b, then there is a point p in pt(L) such that p(a) = 1 and p(b) = 0.
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Patients with type 1 diabetes mellitus require direct injection of insulin as their bodies cannot produce enough (or even any) insulin. As of 2010, there is no other clinically available form of insulin administration other than injection for patients with type 1: injection can be done by insulin pump, by jet injector, or any of several forms of hypodermic needle. Non-injective methods of insulin administration have been unattainable as the insulin protein breaks down in the digestive tract. There are several insulin application mechanisms under experimental development as of 2004, including a capsule that passes to the liver and delivers insulin into the bloodstream.
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The zero object of Ab is the trivial group {0} which consists only of its neutral element. The monomorphisms in Ab are the injective group homomorphisms, the epimorphisms are the surjective group homomorphisms, and the isomorphisms are the bijective group homomorphisms. Ab is a full subcategory of Grp, the category of all groups. The main difference between Ab and Grp is that the sum of two homomorphisms f and g between abelian groups is again a group homomorphism: :(f+g)(x+y) = f(x+y) + g(x+y) = f(x) + f(y) + g(x) + g(y) : = f(x) + g(x) + f(y) + g(y) = (f+g)(x) + (f+g)(y) The third equality requires the group to be abelian.
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Pure subgroups were generalized in several ways in the theory of abelian groups and modules. Pure submodules were defined in a variety of ways, but eventually settled on the modern definition in terms of tensor products or systems of equations; earlier definitions were usually more direct generalizations such as the single equation used above for n'th roots. Pure injective and pure projective modules follow closely from the ideas of Prüfer's 1923 paper. While pure projective modules have not found as many applications as pure injectives, they are more closely related to the original work: A module is pure projective if it is a direct summand of a direct sum of finitely presented modules.
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In complex analysis, a branch of mathematics, Landau's constants are certain mathematical constants that describe the behaviour of holomorphic functions defined on the unit disk. Consider the set F of all those holomorphic functions f on the unit disk for which :f'(0) = 1.\, We define Lf to be the radius of the largest disk contained in the image of f, and Bf to be the radius of the largest disk that is the biholomorphic image of a subset of a unit disk. Landau's constants are then defined as the infimum of Lf or Bf, where f is any holomorphic function or any injective holomorphic function on the unit disk with :f'(0) = 1.
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Let X be an n-dimensional complex projective algebraic variety in CPN, and let Y be a hyperplane section of X such that U = X ∖ Y is smooth. The Lefschetz theorem refers to any of the following statements: # The natural map Hk(Y, Z) → Hk(X, Z) in singular homology is an isomorphism for k < n − 1 and is surjective for k = n − 1\. # The natural map Hk(X, Z) → Hk(Y, Z) in singular cohomology is an isomorphism for k < n − 1 and is injective for k = n − 1\. # The natural map πk(Y, Z) → πk(X, Z) is an isomorphism for k < n − 1 and is surjective for k = n − 1\.
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In the mathematical subject of geometric group theory, a train track map is a continuous map f from a finite connected graph to itself which is a homotopy equivalence and which has particularly nice cancellation properties with respect to iterations. This map sends vertices to vertices and edges to nontrivial edge-paths with the property that for every edge e of the graph and for every positive integer n the path fn(e) is immersed, that is fn(e) is locally injective on e. Train-track maps are a key tool in analyzing the dynamics of automorphisms of finitely generated free groups and in the study of the Culler-Vogtmann Outer space.
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Many properties of morphisms can be restated in terms of points. For example, a map f is said to be a monomorphism if : For all maps g, h, f \circ g = f \circ h implies g = h. Suppose f \colon B \to C and g, h \colon A \to B in C. Then g and h are A-valued points of B, and therefore monomorphism is equivalent to the more familiar statement : f is a monomorphism if it is an injective function on points of B. Some care is necessary. f is an epimorphism if the dual condition holds: : For all maps g, h (of some suitable type), g \circ f = h \circ f implies g = h.
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His abelian category concept had at least partially been anticipated by others. David Buchsbaum in his doctoral thesis written under Eilenberg had introduced a notion of "exact category" close to the abelian category concept (needing only direct sums to be identical); and had formulated the idea of "enough injectives". The Tôhoku paper contains an argument to prove that a Grothendieck category (a particular type of abelian category, the name coming later) has enough injectives; the author indicated that the proof was of a standard type. In showing by this means that categories of sheaves of abelian groups admitted injective resolutions, Grothendieck went beyond the theory available in Cartan–Eilenberg, to prove the existence of a cohomology theory in generality.
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The pigeonhole principle can be extended to infinite sets by phrasing it in terms of cardinal numbers: if the cardinality of set is greater than the cardinality of set , then there is no injection from to . However, in this form the principle is tautological, since the meaning of the statement that the cardinality of set is greater than the cardinality of set is exactly that there is no injective map from to . However, adding at least one element to a finite set is sufficient to ensure that the cardinality increases. Another way to phrase the pigeonhole principle for finite sets is similar to the principle that finite sets are Dedekind finite: Let and be finite sets.
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Let M and N be (left or right) modules over the same ring, and let be a module homomorphism. If M is simple, then f is either the zero homomorphism or injective because the kernel of f is a submodule of M. If N is simple, then f is either the zero homomorphism or surjective because the image of f is a submodule of N. If M = N, then f is an endomorphism of M, and if M is simple, then the prior two statements imply that f is either the zero homomorphism or an isomorphism. Consequently, the endomorphism ring of any simple module is a division ring. This result is known as Schur's lemma.
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The description of the key binary relations has been formulated with the calculus of relations. The univalence property of functions describes a relation R that satisfies the formula R^T R \subseteq I , where I is the identity relation on the range of R. The injective property corresponds to univalence of RT, or the formula R R^T \subseteq I , where this time I is the identity on the domain of R. But a univalent relation is only a partial function, while a univalent total relation is a function. The formula for totality is I \subseteq R R^T . Charles Loewner and Gunther Schmidt use the term mapping for a total, univalent relation.
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Two linked curves forming a Hopf link. When the circle is mapped to three- dimensional Euclidean space by an injective function (a continuous function that does not map two different points of the circle to the same point of space), its image is a closed curve. Two disjoint closed curves that both lie on the same plane are unlinked, and more generally a pair of disjoint closed curves is said to be unlinked when there is a continuous deformation of space that moves them both onto the same plane, without either curve passing through the other or through itself. If there is no such continuous motion, the two curves are said to be linked.
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Since topological indistinguishability is an equivalence relation on any topological space X, we can form the quotient space KX = X/≡. The space KX is called the Kolmogorov quotient or T0 identification of X. The space KX is, in fact, T0 (i.e. all points are topologically distinguishable). Moreover, by the characteristic property of the quotient map any continuous map f : X → Y from X to a T0 space factors through the quotient map q : X → KX. Although the quotient map q is generally not a homeomorphism (since it is not generally injective), it does induce a bijection between the topology on X and the topology on KX. Intuitively, the Kolmogorov quotient does not alter the topology of a space.
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The case n = 2 is less obvious, but can be proven by using basic arguments involving the fundamental groups of the respective spaces: the retraction would induce an injective group homomorphism from the fundamental group of S1 to that of D2, but the first group is isomorphic to Z while the latter group is trivial, so this is impossible. The case n = 2 can also be proven by contradiction based on a theorem about non-vanishing vector fields. For n > 2, however, proving the impossibility of the retraction is more difficult. One way is to make use of homology groups: the homology Hn − 1(Dn) is trivial, while Hn − 1(Sn−1) is infinite cyclic.
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It can be shown that for every marking f the map h : Δk′ → Xn is still injective. The image of h is called the closed simplex in Xn corresponding to f and is denoted by S′(f). Every point in Xn belongs to only finitely many closed simplices and a point of Xn represented by a marking f : Rn → Γ where the graph Γ is tri-valent belongs to a unique closed simplex in Xn, namely S′(f). The weak topology on the Outer space Xn is defined by saying that a subset C of Xn is closed if and only if for every marking f : Rn → Γ the set h−1(C) is closed in Δk′.
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If an abelian category has enough injectives, we can form injective resolutions, i.e. for a given object X we can form a long exact sequence :0\to X \to Q^0 \to Q^1 \to Q^2 \to \cdots and one can then define the derived functors of a given functor F by applying F to this sequence and computing the homology of the resulting (not necessarily exact) sequence. This approach is used to define Ext, and Tor functors and also the various cohomology theories in group theory, algebraic topology and algebraic geometry. The categories being used are typically functor categories or categories of sheaves of OX modules over some ringed space (X, OX) or, more generally, any Grothendieck category.
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Alphasyllabic numeral systems are a type of numeral systems, developed mostly in India starting around 500 AD. Based on various alphasyllabic scripts, in this type of numeral systems glyphs of the numerals are not abstract signs, but syllables of a script, and numerals are represented with these syllable- signs. On the basic principle of these systems, numeric values of the syllables are defined by the consonants and vowels which constitute them, so that consonants and vowels are - or are not in some systems in case of vowels - ordered to numeric values. While there are many hundreds of possible syllables in a script, and since in alphasyllabic numeral systems several syllables receive the same numeric value, so the mapping is not injective.
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In this case, the category is said to be well powered with respect to the class of embeddings. This allows defining new local structures in the category (such as a closure operator). In a concrete category, an embedding is a morphism ƒ: A → B which is an injective function from the underlying set of A to the underlying set of B and is also an initial morphism in the following sense: If g is a function from the underlying set of an object C to the underlying set of A, and if its composition with ƒ is a morphism ƒg: C → B, then g itself is a morphism. A factorization system for a category also gives rise to a notion of embedding.
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The class of sofic groups is closed under the operations of taking subgroups, extensions by amenable groups, and free products. A finitely generated group is sofic if it is the limit of a sequence of sofic groups. The limit of a sequence of amenable groups (that is, an initially subamenable group) is necessarily sofic, but there exist sofic groups that are not initially subamenable groups.. As Gromov proved, Sofic groups are surjunctive. That is, they obey a form of the Garden of Eden theorem for cellular automata defined over the group (dynamical systems whose states are mappings from the group to a finite set and whose state transitions are translation-invariant and continuous) stating that every injective automaton is surjective and therefore also reversible.
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Mapping cylinders are quite common homotopical tools. One use of mapping cylinders is to apply theorems concerning inclusions of spaces to general maps, which might not be injective. Consequently, theorems or techniques (such as homology, cohomology or homotopy theory) which are only dependent on the homotopy class of spaces and maps involved may be applied to f\colon X\rightarrow Y with the assumption that X \subset Y and that f is actually the inclusion of a subspace. Another, more intuitive appeal of the construction is that it accords with the usual mental image of a function as "sending" points of X to points of Y, and hence of embedding X within Y, despite the fact that the function need not be one-to-one.
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To be the transition function of a cellular automaton, a function from states to states must be a continuous function for this topology, and must also be equivariant with the group action, meaning that shifting the cells prior to applying the transition function produces the same result as applying the function and then shifting the cells. For such functions, the Curtis–Hedlund–Lyndon theorem ensures that the value of the transition function at each group element depends on the previous state of only a finite set of neighboring elements. A state transition function is a surjective function when every state has a predecessor (there can be no Garden of Eden). It is an injective function when no two states have the same successor.
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This is often denoted as just: : The above definition is extended to directed graphs. Then, for a homomorphism f : G → H, (f(u),f(v)) is an arc (directed edge) of H whenever (u,v) is an arc of G. There is an injective homomorphism from G to H (i.e., one that never maps distinct vertices to one vertex) if and only if G is a subgraph of H. If a homomorphism f : G → H is a bijection (a one-to-one correspondence between vertices of G and H) whose inverse function is also a graph homomorphism, then f is a graph isomorphism. Covering maps are a special kind of homomorphisms that mirror the definition and many properties of covering maps in topology.
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In the category of sets, every monomorphism (injective function) with a non-empty domain is a section, and every epimorphism (surjective function) is a retraction; the latter statement is equivalent to the axiom of choice. In the category of vector spaces over a field K, every monomorphism and every epimorphism splits; this follows from the fact that linear maps can be uniquely defined by specifying their values on a basis. In the category of abelian groups, the epimorphism Z → Z/2Z which sends every integer to its remainder modulo 2 does not split; in fact the only morphism Z/2Z → Z is the zero map. Similarly, the natural monomorphism Z/2Z → Z/4Z doesn't split even though there is a non-trivial morphism Z/4Z → Z/2Z.
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In abstract algebra, the Eakin–Nagata theorem states: given commutative rings A \subset B such that B is finitely generated as a module over A, if B is a Noetherian ring, then A is a Noetherian ring. (Note the converse is also true and is easier.) The theorem is similar to the Artin–Tate lemma, which says that the same statement holds with "Noetherian" replaced by "finitely generated algebra" (assuming the base ring is a Noetherian ring). The theorem was first proved in Paul M. Eakin's thesis and later independently by . The theorem can also be deduced from the characterization of a Noetherian ring in terms of injective modules, as done for example by David Eisenbud in ; this approach is useful for a generalization to non-commutative rings.
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But it turns out that (if A is "nice" enough) there is one canonical way of doing so, given by the right derived functors of F. For every i≥1, there is a functor RiF: A → B, and the above sequence continues like so: 0 → F(A) → F(B) → F(C) → R1F(A) → R1F(B) → R1F(C) → R2F(A) → R2F(B) → ... . From this we see that F is an exact functor if and only if R1F = 0; so in a sense the right derived functors of F measure "how far" F is from being exact. If the object A in the above short exact sequence is injective, then the sequence splits. Applying any additive functor to a split sequence results in a split sequence, so in particular R1F(A) = 0.
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In von Neumann's set-theoretic construction of the natural numbers, the number 1 is defined as the singleton {0}. In axiomatic set theory, the existence of singletons is a consequence of the axiom of pairing: for any set A, the axiom applied to A and A asserts the existence of {A, A}, which is the same as the singleton {A} (since it contains A, and no other set, as an element). If A is any set and S is any singleton, then there exists precisely one function from A to S, the function sending every element of A to the single element of S. Thus every singleton is a terminal object in the category of sets. A singleton has the property that every function from it to any arbitrary set is injective.
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This does not depend upon the projectivity of P: it is true of all superfluous epimorphisms. If (P,p) is a projective cover of M, and P' is another projective module with an epimorphism p':P'\rightarrow M, then there is a split epimorphism α from P' to P such that p\alpha=p' Unlike injective envelopes and flat covers, which exist for every left (right) R-module regardless of the ring R, left (right) R-modules do not in general have projective covers. A ring R is called left (right) perfect if every left (right) R-module has a projective cover in R-Mod (Mod-R). A ring is called semiperfect if every finitely generated left (right) R-module has a projective cover in R-Mod (Mod-R).
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The inverse of this restriction can be extended uniquely to a ring homomorphism φn from R[X1,...,Xn]Sn to R[X1,...,Xn+1]Sn+1, as follows for instance from the fundamental theorem of symmetric polynomials. Since the images φn(ek(X1,...,Xn)) = ek(X1,...,Xn+1) for k = 1,...,n are still algebraically independent over R, the homomorphism φn is injective and can be viewed as a (somewhat unusual) inclusion of rings; applying φn to a polynomial amounts to adding all monomials containing the new indeterminate obtained by symmetry from monomials already present. The ring ΛR is then the "union" (direct limit) of all these rings subject to these inclusions. Since all φn are compatible with the grading by total degree of the rings involved, ΛR obtains the structure of a graded ring.
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In a more quantified version: for natural numbers k and m, if n = km + 1 objects are distributed among m sets, then the pigeonhole principle asserts that at least one of the sets will contain at least k+1 objects. For arbitrary n and m this generalizes to k + 1 = \lfloor(n - 1)/m \rfloor + 1 = \lceil n/m\rceil, where \lfloor\cdots\rfloor and \lceil\cdots\rceil denote the floor and ceiling functions, respectively. Though the most straightforward application is to finite sets (such as pigeons and boxes), it is also used with infinite sets that cannot be put into one-to-one correspondence. To do so requires the formal statement of the pigeonhole principle, which is "there does not exist an injective function whose codomain is smaller than its domain".
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Obviously, if the mapping fails to be injective, at some point in the simulation (for example as the last step) some "garbage" has to be produced. For quantum circuits a similar composition of qubit gates can be defined. That is, associated to any classical assemblage as above, we can produce a reversible quantum circuit when in place of f we have an n-qubit gate U and in place of g we have an m-qubit gate W. See illustration below: 300px The fact that connecting gates this way gives rise to a unitary mapping on n+m−k qubit space is easy to check. In a real quantum computer the physical connection between the gates is a major engineering challenge, since it is one of the places where decoherence may occur.
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If distinguishes points of and if denotes the range of the injection then is a vector subspace of the algebraic dual space of and the pairing becomes canonically identified with the canonical pairing (where is the natural evaluation map). In particular, in this situation we can assume without loss of generality that is a vector subspace of 's algebraic dual and is the evaluation map. :Convention: Often, whenever is injective (especially when forms a dual pair) then we will use the common practice of assuming without loss of generality that is a vector subspace of the algebraic dual space of , that is the natural evaluation map, and we may also denote by . In a completely analogous manner, if distinguishes points of then it is possible for to be identified as a vector subspace of 's algebraic dual space.
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However, if M3 is instead embedded into a metric space that contains a fourth point at distance 1/2 from each of the three points of M3, the Čech complex of the radius-1/2 balls in this space would contain the triangle. Thus, the Čech complex of fixed-radius balls centered at M3 differs depending on which larger space M3 might be embedded into, while the Vietoris–Rips complex remains unchanged. If any metric space X is embedded in an injective metric space Y, the Vietoris–Rips complex for distance δ and X coincides with the Čech complex of the balls of radius δ/2 centered at the points of X in Y. Thus, the Vietoris–Rips complex of any metric space M equals the Čech complex of a system of balls in the tight span of M.
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If ƒ must be injective, then the selection must involve n distinct elements of X, so it is a subset of X of size n, also called an n-combination. Without the requirement, a same element of X may occur multiple times in the selection, and the result is a multiset of size n of elements from X, also called an n-multicombination or n-combination with repetition. In these cases the requirement of a surjective ƒ means that every label is to be used at least once, respectively that every element of X be included in the selection at least once. Such a requirement is less natural to handle mathematically, and indeed the former case is more easily viewed first as a grouping of elements of N, with in addition a labelling of the groups by the elements of X.
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The usual category theoretical definition is in terms of the property of lifting that carries over from free to projective modules: a module P is projective if and only if for every surjective module homomorphism and every module homomorphism , there exists a module homomorphism such that . (We don't require the lifting homomorphism h to be unique; this is not a universal property.) :120px The advantage of this definition of "projective" is that it can be carried out in categories more general than module categories: we don't need a notion of "free object". It can also be dualized, leading to injective modules. The lifting property may also be rephrased as every morphism from P to M factors through every epimorphism to M. Thus, by definition, projective modules are precisely the projective objects in the category of R-modules.
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In the case where a graph H can be obtained from a graph G by a sequence of lifting operations (on G) and then finding an isomorphic subgraph, we say that H is an immersion minor of G. There is yet another way of defining immersion minors, which is equivalent to the lifting operation. We say that H is an immersion minor of G if there exists an injective mapping from vertices in H to vertices in G where the images of adjacent elements of H are connected in G by edge- disjoint paths. The immersion minor relation is a well-quasi-ordering on the set of finite graphs and hence the result of Robertson and Seymour applies to immersion minors. This furthermore means that every immersion minor-closed family is characterized by a finite family of forbidden immersion minors.
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A map f: A \to X is called a cofibration if given (1) a map h_0 : X \to Z and (2) a homotopy g_t : A \to Z, there exists a homotopy h_t : X \to Z that extends h_0 and such that h_t \circ f = g_t. To some loose sense, it is an analog of the defining diagram of an injective module in abstract algebra. The most basic example is a CW pair (X, A); since many work only with CW complexes, the notion of a cofibration is often implicit. A fibration in the sense of Serre is the dual notion of a cofibration: that is, a map p : X \to B is a fibration if given (1) a map Z \to X and (2) a homotopy g_t : Z \to B, there exists a homotopy h_t: Z \to X such that h_0 is the given one and p \circ h_t = g_t.
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Equivalently, the inverse linear operator (T − λ)−1, which is defined on the dense subset R, is not a bounded operator, and therefore cannot be extended to the whole of X. Then λ is said to be in the continuous spectrum, σc(T), of T. #T − λ is injective but does not have dense range. That is, there is some element x in X and a neighborhood N of x such that (T − λ)(y) is never in N. In this case, the map (T − λ)−1 x → x may be bounded or unbounded, but in any case does not admit a unique extension to a bounded linear map on all of X. Then λ is said to be in the residual spectrum of T, σr(T). So σ(T) is the disjoint union of these three sets, :\sigma(T) = \sigma_p (T) \cup \sigma_c (T) \cup \sigma_r (T).
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In universal algebra, an abstract algebra A is called simple if and only if it has no nontrivial congruence relations, or equivalently, if every homomorphism with domain A is either injective or constant. As congruences on rings are characterized by their ideals, this notion is a straightforward generalization of the notion from ring theory: a ring is simple in the sense that it has no nontrivial ideals if and only if it is simple in the sense of universal algebra. The same remark applies with respect to groups and normal subgroups; hence the universal notion is also a generalization of a simple group (it is a matter of convention whether a one-element algebra should be or should not be considered simple, hence only in this special case the notions might not match). A theorem by Roberto Magari in 1969 asserts that every variety contains a simple algebra.
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For any bounded set S in any metric space, d/2 ≤ r ≤ d. The first inequality is implied by the triangle inequality for the center of the ball and the two diametral points, and the second inequality follows since a ball of radius d centered at any point of S will contain all of S. In a uniform metric space, that is, a space in which all distances are equal, r = d. At the other end of the spectrum, in an injective metric space such as the Manhattan distance in the plane, r = d/2: any two closed balls of radius d/2 centered at points of S have a nonempty intersection, therefore all such balls have a common intersection, and a radius d/2 ball centered at a point of this intersection contains all of S. Versions of Jung's theorem for various non-Euclidean geometries are also known (see e.g. Dekster 1995, 1997).
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The figure shows a set X of 16 points in the plane; to form a finite metric space from these points, we use the Manhattan distance (L1 metric).In two dimensions, the Manhattan distance is isometric after rotation and scaling to the L∞ distance, so with this metric the plane is itself injective, but this equivalence between L1 and L∞ does not hold in higher dimensions. The blue region shown in the figure is the orthogonal convex hull, the set of points z such that each of the four closed quadrants with z as apex contains a point of X. Any such point z corresponds to a point of the tight span: the function f(x) corresponding to a point z is f(x) = d(z,x). A function of this form satisfies property 1 of the tight span for any z in the Manhattan-metric plane, by the triangle inequality for the Manhattan metric.
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Traditionally many of the problems in the twelvefold way have been formulated in terms of placing balls in boxes (or some similar visualization) instead of defining functions. The set N can be identified with a set of balls, and X with a set of boxes; the function ƒ : then describes a way to distribute the balls into the boxes, namely by putting each ball a into box ƒ(a). Thus the property that a function ascribes a unique image to each value in its domain is reflected by the property that any ball can go into only one box (together with the requirement that no ball should remain outside of the boxes), whereas any box can accommodate (in principle) an arbitrary number of balls. Requiring in addition ƒ to be injective means forbidding to put more than one ball in any one box, while requiring ƒ to be surjective means insisting that every box contain at least one ball.
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Instead of just one elliptic operator, one can consider a family of elliptic operators parameterized by some space Y. In this case the index is an element of the K-theory of Y, rather than an integer. If the operators in the family are real, then the index lies in the real K-theory of Y. This gives a little extra information, as the map from the real K theory of Y to the complex K theory is not always injective. Atiyah's former student Graeme Segal (in 1982), who worked with Atiyah on equivariant K-theory With Bott, Atiyah found an analogue of the Lefschetz fixed-point formula for elliptic operators, giving the Lefschetz number of an endomorphism of an elliptic complex in terms of a sum over the fixed points of the endomorphism. As special cases their formula included the Weyl character formula, and several new results about elliptic curves with complex multiplication, some of which were initially disbelieved by experts.
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A presheaf F on a topological space is called a sheaf if it satisfies the sheaf condition: whenever an open subset is covered by open subsets Ui, and we are given elements of F(Ui) for all i whose restrictions to Ui ∩ Uj agree for all i, j, then they are images of a unique element of F(U). By analogy, an étale presheaf is called a sheaf if it satisfies the same condition (with intersections of open sets replaced by pullbacks of étale morphisms, and where a set of étale maps to U is said to cover U if the topological space underlying U is the union of their images). More generally, one can define a sheaf for any Grothendieck topology on a category in a similar way. The category of sheaves of abelian groups over a scheme has enough injective objects, so one can define right derived functors of left exact functors.
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Alternatively, one may require that the pairing be a nondegenerate form, meaning that for all non-zero there exists some such that , though need not equal ; in other words, the induced map to the dual space is injective. This generalization is important in differential geometry: a manifold whose tangent spaces have an inner product is a Riemannian manifold, while if this is related to nondegenerate conjugate symmetric form the manifold is a pseudo- Riemannian manifold. By Sylvester's law of inertia, just as every inner product is similar to the dot product with positive weights on a set of vectors, every nondegenerate conjugate symmetric form is similar to the dot product with nonzero weights on a set of vectors, and the number of positive and negative weights are called respectively the positive index and negative index. Product of vectors in Minkowski space is an example of indefinite inner product, although, technically speaking, it is not an inner product according to the standard definition above.
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A group G is said to be linear if there exists a field K, an integer d and an injective morphism from G to the general linear group GLd(K) (a faithful linear representation of dimension d over K): if needed one can mention the field and dimension by saying that G is linear of degree d over K. Basic instances are groups which are defined as subgroups of a linear group, for example: #The group GLn(K) itself; #The special linear group SLn(K) (the subgroup of matrices with determinant 1); #The group of invertible upper (or lower) triangular matrices #If gi is a collection of elements in GLn(K) indexed by a set I, then the subgroup generated by the gi is a linear group. In the study of Lie groups, it is sometimes pedagogically convenient to restrict attention to Lie groups that can be faithfully represented over the field of complex numbers. (Some authors require that the group be represented as a closed subgroup of the GLn(C).) Books that follow this approach include Hall (2015) and Rossman (2002).
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A motion is a loop in the configuration space, which consists of all possible ways of embedding n circles into the 3-disk. This becomes a group in the same way as loops in any space can be made into a group; first, we define equivalence classes of loops by letting paths g and h be equivalent iff they are related by a (smooth) homotopy, and then we define a group operation on the equivalence classes by concatenation of paths. In his 1962 Ph.D. thesis, David M. Dahm was able to show that there is an injective homomorphism from this group into the automorphism group of the free group on n generators, so it is natural to identify the group with this subgroup of the automorphism group.. One may also show that the loop braid group is isomorphic to the welded braid group, as is done for example in a paper by John C. Baez, Derek Wise, and Alissa Crans, which also gives some presentations of the loop braid group using the work of Xiao-Song Lin..
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This group is unique in the sense that every two free abelian groups with the same basis are isomorphic. Instead of constructing it by describing its individual elements, a free group with basis B may be constructed as a direct sum of copies of the additive group of the integers, with one copy per member of B. Alternatively, the free abelian group with basis B may be described by a presentation with the elements of B as its generators and with the commutators of pairs of members as its relators. The rank of a free abelian group is the cardinality of a basis; every two bases for the same group give the same rank, and every two free abelian groups with the same rank are isomorphic. Every subgroup of a free abelian group is itself free abelian; this fact allows a general abelian group to be understood as a quotient of a free abelian group by "relations", or as a cokernel of an injective homomorphism between free abelian groups.
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For a pairing between X and Y (where Y need not be different from X) to be a bijection, four properties must hold: # each element of X must be paired with at least one element of Y, # no element of X may be paired with more than one element of Y, # each element of Y must be paired with at least one element of X, and # no element of Y may be paired with more than one element of X. Satisfying properties (1) and (2) means that a pairing is a function with domain X. It is more common to see properties (1) and (2) written as a single statement: Every element of X is paired with exactly one element of Y. Functions which satisfy property (3) are said to be "onto Y " and are called surjections (or surjective functions). Functions which satisfy property (4) are said to be "one-to-one functions" and are called injections (or injective functions).There are names associated to properties (1) and (2) as well. A relation which satisfies property (1) is called a total relation and a relation satisfying (2) is a single valued relation.
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Skew tableau of shape (5, 4, 2, 2) / (2, 1), English notation A skew shape is a pair of partitions (, ) such that the Young diagram of contains the Young diagram of ; it is denoted by . If and , then the containment of diagrams means that for all . The skew diagram of a skew shape is the set-theoretic difference of the Young diagrams of and : the set of squares that belong to the diagram of but not to that of . A skew tableau of shape is obtained by filling the squares of the corresponding skew diagram; such a tableau is semistandard if entries increase weakly along each row, and increase strictly down each column, and it is standard if moreover all numbers from 1 to the number of squares of the skew diagram occur exactly once. While the map from partitions to their Young diagrams is injective, this is not the case from the map from skew shapes to skew diagrams;For instance the skew diagram consisting of a single square at position (2,4) can be obtained by removing the diagram of from the one of , but also in (infinitely) many other ways.
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