And if each linearization is only surjective, and a family of right inverses is smooth tame, then P is locally surjective with a smooth tame right inverse.
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A ', or a ', which is invariant under surjective endomorphisms. For finite groups, surjectivity of an endomorphism implies injectivity, so a surjective endomorphism is an automorphism; thus being strictly characteristic is equivalent to characteristic. This is not the case anymore for infinite groups.
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The continuous function in this theorem is not required to be bijective or even surjective.
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Then K is normal in FG, therefore is equal to its normal closure, so . Since the identity map is surjective, φ is also surjective, so by the First Isomorphism Theorem, . This presentation may be highly inefficient if both G and K are much larger than necessary. > Corollary.
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If the image of is non-meager in then is a surjective open map and is a complete metrizable space.
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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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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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Alternative (equivalent) formulations of the definition in terms of a bijective function or a surjective function can also be given. See below.
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In this section, we give two different constructions of a two-to-one and surjective homomorphism of SU(2) onto SO(3).
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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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Both transitions are not surjective, that is, not every B-space results from some A-space. First, a 3-dim Euclidean space is a special (not general) case of a Euclidean space. Second, a topology of a Euclidean space is a special case of topology (for instance, it must be non-compact, and connected, etc). We denote surjective transitions by a two-headed arrow, "↠" rather than "→".
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It is surjective, and therefore an isomorphism, if and only if the dimension of is finite. This fact characterizes finite-dimensional vector spaces without referring to a basis.
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A triple where and are differentiable manifolds and is a surjective submersion, is called a fibered manifold. E is called the total space, B is called the base.
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It is clear that K1 and K2 are invariant subspaces of V. So V(K2) = K2. In other words, V restricted to K2 is a surjective isometry, i.e.
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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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In algebra, epimorphisms are often defined as surjective homomorphisms. On the other hand, in category theory, epimorphisms are defined as right cancelable morphisms. This means that a (homo)morphism f: A \to B is an epimorphism if, for any pair g, h of morphisms from B to any other object C, the equality g \circ f = h \circ f implies g = h. A surjective homomorphism is always right cancelable, but the converse is not always true for algebraic structures.
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The importance of these structures in all mathematics, and specially in linear algebra and homological algebra, may explain the coexistence of two non-equivalent definitions. Algebraic structures for which there exist non- surjective epimorphisms include semigroups and rings. The most basic example is the inclusion of integers into rational numbers, which is an homomorphism of rings and of multiplicative semigroups. For both structures it is a monomorphism and a non-surjective epimorphism, but not an isomorphism.
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If X is a scheme of finite type over a field there is a natural map from divisors to line bundles. If X is either projective or reduced then this map is surjective. Kleiman found an example of a non-reduced and non-projective X for which this map is not surjective as follows. Take Hironaka's example of a variety with two rational curves A and B such that A+B is numerically equivalent to 0.
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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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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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A solution (sometimes also called weak solution) of such an embedding problem is a continuous homomorphism γ : F → H such that φ = f γ. If the solution is surjective, it is called a proper solution.
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In mathematics, in particular, in algebraic geometry, an isogeny is a morphism of algebraic groups (a.k.a group varieties) that is surjective and has a finite kernel. If the groups are abelian varieties, then any morphism f : A → B of the underlying algebraic varieties which is surjective with finite fibres is automatically an isogeny, provided that f(1A) = 1B. Such an isogeny f then provides a group homomorphism between the groups of k-valued points of A and B, for any field k over which f is defined.
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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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In differential geometry, a fibered manifold is surjective submersion of smooth manifolds . Locally trivial fibered manifolds are fiber bundles. Therefore, a notion of connection on fibered manifolds provides a general framework of a connection on fiber bundles.
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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 submersion is a differentiable map between differentiable manifolds whose differential is everywhere surjective. This is a basic concept in differential topology. The notion of a submersion is dual to the notion of an immersion.
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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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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 inverse of a bijective continuous map is a bijective open/closed map (and vice versa). A surjective open map is not necessarily a closed map, and likewise, a surjective closed map is not necessarily an open map. A variant of the closed map lemma states that if a continuous function between locally compact Hausdorff spaces is proper, then it is also closed. In complex analysis, the identically named open mapping theorem states that every non-constant holomorphic function defined on a connected open subset of the complex plane is an open map.
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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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However, the two definitions of epimorphism are equivalent for sets, vector spaces, abelian groups, modules (see below for a proof), and groups.Linderholm, C. E. (1970). A group epimorphism is surjective. The American Mathematical Monthly, 77(2), 176-177.
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This difference is, however, the only one: if X is path-connected, this homomorphism is surjective and its kernel is the commutator subgroup of the fundamental group, so that H_1(X) is isomorphic to the abelianization of the fundamental group.
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In algebraic geometry, the syntomic topology is a Grothendieck topology introduced by . Mazur defined a morphism to be syntomic if it is flat and locally a complete intersection. The syntomic topology is generated by surjective syntomic morphisms of affine schemes.
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The theorem on the surjection of Fréchet spaces is an important theorem, due to Stefan Banach, that characterizes when a continuous linear operator between Fréchet spaces is surjective. The importance of this theorem is related to the open mapping theorem, which states that a continuous linear surjection between Fréchet spaces is an open map. Often in practice, one knows that they have a continuous linear map between Fréchet spaces and wishes to show that it is surjective in order to use the open mapping theorem to deduce that it is also an open mapping. This theorem may help reach that goal.
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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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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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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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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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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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Let X and Y be topological spaces and let p be a map from X to Y that is continuous, closed, surjective and such that p^{-1}(y) is compact relative to X for each y in Y. Then p is known as a perfect map.
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Every perfect map is a quotient map. This follows from the fact that a closed, continuous surjective map is always a quotient map. 11\. Let G be a compact topological group which acts continuously on X. Then the quotient map from X to X/G is a perfect map.
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The homomorphism F→M is defined to be a flat cover of M if it is surjective, F is flat, every homomorphism from flat module to M factors through F, and any map from F to F commuting with the map to M is an automorphism of F.
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A map is called a quasi-isometric embedding if it satisfies the first condition but not necessarily the second (i.e. it is coarsely Lipschitz but may fail to be coarsely surjective). In other words, if through the map, (M_1,d_1) is quasi-isometric to a subspace of (M_2,d_2).
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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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One can identify N as a normal subgroup of G, and N' as a normal subgroup of G'. Then the image of H in G/N\times G'/N' is the graph of an isomorphism G/N\approx G'/N'. An immediate consequence of this is that the subdirect product of two groups can be described as a fiber product and vice versa. Notice that if H is any subgroup of G\times G' (the projections p_1: H\rightarrow G and p_2: H\rightarrow G' need not be surjective), then the projections from H onto p_1(H) and p_2(H) are surjective. Then one can apply Goursat's lemma to H \leq p_1(H)\times p_2(H).
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Equivalently, it is a surjective TVS embedding. Many properties of TVSs that are studied, such as local convexity, metrizability, completeness, and normability, are invariant under TVS isomorphisms. ;A necessary condition for a vector topology All of the above conditions are consequently a necessity for a topology to form a vector topology.
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Note that if a linear operator T : X → Y is almost open then because T(X) is a vector subspace of Y that contains a neighborhood of 0 in Y, T : X → Y is necessarily surjective. For this reason many authors require surjectivity as part of the definition of "almost open".
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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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Let h\colon B\to C be the zero map. If f is not surjective, C eq 0, and thus g eq h (one is a zero map, while the other is not). Thus f is not cancelable, as g \circ f = h \circ f (both are the zero map from A to C).
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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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There is a natural quotient functor from C to C/R which sends each morphism to its equivalence class. This functor is bijective on objects and surjective on Hom- sets (i.e. it is a full functor). Every functor F : C → D determines a congruence on C by saying f ~ g iff F(f) = F(g).
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An order-isomorphism is a monotone bijective function that has a monotone inverse. This is equivalent to being a surjective order-embedding. Hence, the image f(P) of an order-embedding is always isomorphic to P, which justifies the term "embedding". A more elaborate type of functions is given by so-called Galois connections.
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In a different direction, it is finer than the qfh topology, so h locally, algebraic correspondences are finite sums of morphisms.Suslin, Voevodsky, Singular homology of abstract algebraic varieties Finally, every proper surjective morphism is an h covering, so in any situation where de Jong's theorem on alterations is valid, h locally all schemes are regular.
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Let X be a compact Kähler manifold. The first Chern class c1 gives a map from holomorphic line bundles to . By Hodge theory, the de Rham cohomology group H2(X, C) decomposes as a direct sum , and it can be proven that the image of c1 lies in H1,1(X). The theorem says that the map to is surjective.
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A serial relation can be equivalently characterized as every element having a non-empty successor neighborhood. Similarly an inverse serial relation is a relation in which every element has non-empty "predecessor neighborhood". More commonly, an inverse serial relation is called a surjective relation, and is specified by a serial converse relation. Definition 5.8, page 57.
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But if the dimension of is less than the dimension of , all points are critical according to the definition above (the differential cannot be surjective) but the rank of the Jacobian may still be maximal (if it is equal to dim ). The definition given above is the more commonly used; e.g., in the formulation of Sard's theorem.
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A linear functional is non-trivial if and only if it is surjective (i.e. its range is all of ).This follows since just as the image of a vector subspace under a linear transformation is a vector subspace, so is the image of under . However, the only vector subspaces (that is, -subspaces) of are } and itself.
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179, Belmont: Wadsworth More generally, the term motion is a synonym for surjective isometry in metric geometry,M.A. Khamsi & W.A. Kirk (2001) An Introduction to Metric Spaces and Fixed Point Theorems, p. 15, John Wiley & Sons including elliptic geometry and hyperbolic geometry. In the latter case, hyperbolic motions provide an approach to the subject for beginners.
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Elman, Karpenko, Merkurjev (2008), section 5. The homomorphism is surjective, since the Pfister forms additively generate In. One part of the Milnor conjecture, proved by Orlov, Vishik and Voevodsky, states that this homomorphism is in fact an isomorphism kn(F) ≅ In/In+1.Orlov, Vishik, Voevodsky (2007). That gives an explicit description of the abelian group In/In+1 by generators and relations.
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Let R and S be rings, and let φ: R → S be a ring homomorphism. Then: # The kernel of φ is an ideal of R, # The image of φ is a subring of S, and # The image of φ is isomorphic to the quotient ring R / ker(φ). In particular, if φ is surjective then S is isomorphic to R / ker(φ).
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Let M and N be modules, and let φ: M → N be a module homomorphism. Then: # The kernel of φ is a submodule of M, # The image of φ is a submodule of N, and # The image of φ is isomorphic to the quotient module M / ker(φ). In particular, if φ is surjective then N is isomorphic to M / ker(φ).
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Let X be a discrete topological space, and let \omega X be an Alexandroff one-point compactification of X. A Hausdorff space P is polyadic if for some cardinal number \lambda, there exists a continuous surjective function f : \omega X^\lambda \rightarrow P, where \omega X^\lambda is the product space obtained by multiplying \omega X with itself \lambda times.
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Goursat's lemma for groups can be stated as follows. :Let G, G' be groups, and let H be a subgroup of G\times G' such that the two projections p_1: H\rightarrow G and p_2: H\rightarrow G' are surjective (i.e., H is a subdirect product of G and G'). Let N be the kernel of p_2 and N' the kernel of p_1.
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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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In algebra, the congruence ideal of a surjective ring homomorphism f : B → C of commutative rings is the image under f of the annihilator of the kernel of f. It is called a congruence ideal because when B is a Hecke algebra and f is a homomorphism corresponding to a modular form, the congruence ideal describes congruences between the modular form of f and other modular forms.
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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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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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In mathematics, an index set is a set whose members label (or index) members of another set. For instance, if the elements of a set may be indexed or labeled by means of the elements of a set , then is an index set. The indexing consists of a surjective function from onto , and the indexed collection is typically called an (indexed) family, often written as .
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These sizes moreover do not come in any definite order, while the same size may occur more than once; one may choose to arrange them into a weakly decreasing list of numbers, whose sum is the number n. This gives the combinatorial notion of a partition of the number n, into exactly x (for surjective ƒ) or at most x (for arbitrary ƒ) parts.
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The companion terms epimorphism and monomorphism were first introduced by Bourbaki. Bourbaki uses epimorphism as shorthand for a surjective function. Early category theorists believed that epimorphisms were the correct analogue of surjections in an arbitrary category, similar to how monomorphisms are very nearly an exact analogue of injections. Unfortunately this is incorrect; strong or regular epimorphisms behave much more closely to surjections than ordinary epimorphisms.
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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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Such a map is always surjective and has a finite kernel, the order of which is the degree of the isogeny. Points on correspond to pairs of elliptic curves admitting an isogeny of degree with cyclic kernel. When has genus one, it will itself be isomorphic to an elliptic curve, which will have the same -invariant. For instance, has -invariant , and is isomorphic to the curve .
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In Riemannian geometry, the smooth coarea formulas relate integrals over the domain of certain mappings with integrals over their codomains. Let \scriptstyle M,\,N be smooth Riemannian manifolds of respective dimensions \scriptstyle m\,\geq\, n. Let \scriptstyle F:M\,\longrightarrow\, N be a smooth surjection such that the pushforward (differential) of \scriptstyle F is surjective almost everywhere. Let \scriptstyle\varphi:M\,\longrightarrow\, [0,\infty) a measurable function.
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Thus, the matrix exponential of a Hamiltonian matrix is symplectic. However the logarithm of a symplectic matrix is not necessarily Hamiltonian because the exponential map from the Lie algebra to the group is not surjective.. The characteristic polynomial of a real Hamiltonian matrix is even. Thus, if a Hamiltonian matrix has as an eigenvalue, then , and are also eigenvalues. It follows that the trace of a Hamiltonian matrix is zero.
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Diagram of the fundamental theorem on homomorphisms Let G and H be groups, and let φ: G → H be a homomorphism. Then: # The kernel of φ is a normal subgroup of G, # The image of φ is a subgroup of H, and # The image of φ is isomorphic to the quotient group G / ker(φ). In particular, if φ is surjective then H is isomorphic to G / ker(φ).
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This implies that p is an irreducible polynomial, and thus that the quotient ring K[X]/\langle p \rangle is a field. As L is generated by θ, \varphi is surjective, and \varphi induces an isomorphism from K[X]/\langle p \rangle onto L. This implies that every element of L is equal to a unique polynomial in θ, of degree lower than the degree of the extension.
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In the context of frames and locales, the composite is called the nucleus induced by . Nuclei induce frame homomorphisms; a subset of a locale is called a sublocale if it is given by a nucleus. Conversely, any closure operator on some poset gives rise to the Galois connection with lower adjoint being just the corestriction of to the image of (i.e. as a surjective mapping the closure system ).
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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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In mathematics, the Federer–Morse theorem, introduced by , states that if f is a surjective continuous map from a compact metric space X to a compact metric space Y, then there is a Borel subset Z of X such that f restricted to Z is a bijection from Z to Y. Moreover, the inverse of that restriction is a Borel section of f - it is a Borel isomorphism.
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Let A be a linear operator defined on a linear subspace D(A) of the Banach space X. Then A generates a contraction semigroup if and only ifEngel and Nagel Theorem II.3.15, Arent et al. Theorem 3.4.5, Staffans Theorem 3.4.8 # D(A) is dense in X, # A is closed, # A is dissipative, and # A − λ0I is surjective for some λ0> 0, where I denotes the identity operator.
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Let A be a linear operator defined on a linear subspace D(A) of the Banach space X. Then A generates a quasi contraction semigroup if and only if # D(A) is dense in X, # A is closed, # A is quasidissipative, i.e. there exists a ω ≥ 0 such that A − ωI is dissipative, and # A − λ0I is surjective for some λ0 > ω, where I denotes the identity operator.
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Under this convention, all functions are surjective, so bijectivity and injectivity are the same. Authors using this convention may use the phrasing that a function is invertible if and only if it is an injection. The two conventions need not cause confusion, as long as it is remembered that in this alternate convention, the codomain of a function is always taken to be the image of the function.
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In abstract algebra, one uses homology to define derived functors, for example the Tor functors. Here one starts with some covariant additive functor F and some module X. The chain complex for X is defined as follows: first find a free module F1 and a surjective homomorphism p1 : F1 → X. Then one finds a free module F2 and a surjective homomorphism p2 : F2 → ker(p1). Continuing in this fashion, a sequence of free modules Fn and homomorphisms pn can be defined. By applying the functor F to this sequence, one obtains a chain complex; the homology Hn of this complex depends only on F and X and is, by definition, the n-th derived functor of F, applied to X. A common use of group (co)homology H^2(G,M)is to classify the possible extension groups E which contain a given G-module M as a normal subgroup and have a given quotient group G, so that G = E/M.
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This Whitehead/Russell theorem is described in more modern language by . # Every surjective function from P(P(S)) onto itself is one-to-one. # (Alfred Tarski) Every non-empty family of subsets of S has a minimal element with respect to inclusion., demonstrated that his definition (which is also known as I-finite) is equivalent to Kuratowski's set-theoretical definition, which he then noted is equivalent to the standard numerical definition via the proof by .
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Note \pi need not be surjective. Also, if it exists, a categorical quotient is unique up to a canonical isomorphism. In practice, one takes C to be the category of varieties or the category of schemes over a fixed scheme. A categorical quotient \pi is a universal categorical quotient if it is stable under base change: for any Y' \to Y, \pi': X' = X \times_Y Y' \to Y' is a categorical quotient.
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Saunders Mac Lane attempted to create a distinction between epimorphisms, which were maps in a concrete category whose underlying set maps were surjective, and epic morphisms, which are epimorphisms in the modern sense. However, this distinction never caught on. It is a common mistake to believe that epimorphisms are either identical to surjections or that they are a better concept. Unfortunately this is rarely the case; epimorphisms can be very mysterious and have unexpected behavior.
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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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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 abstract algebra, a cover is one instance of some mathematical structure mapping onto another instance, such as a group (trivially) covering a subgroup. This should not be confused with the concept of a cover in topology. When some object X is said to cover another object Y, the cover is given by some surjective 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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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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This map is closed, continuous (by the pasting lemma), and surjective and therefore is a perfect map (the other condition is trivially satisfied). However, p is not open, for the image of under p is which is not open relative to (the range of p). Note that this map is a quotient map and the quotient operation is 'gluing' two intervals together. 8\. Notice how, to preserve properties such as local connectedness, second countability, local compactness etc.
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A wide generalization of this example is the localization of a ring by a multiplicative set. Every localization is a ring epimorphism, which is not, in general, surjective. As localizations are fundamental in commutative algebra and algebraic geometry, this may explain why in these areas, the definition of epimorphisms as right cancelable homomorphisms is generally preferred. A split epimorphism is a homomorphism that has a right inverse and thus it is itself a left inverse of that other homomorphism.
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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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In mathematics, or specifically, in differential topology, Ehresmann's lemma or Ehresmann's fibration theorem states that a smooth mapping f\colon M \rightarrow N where M and N are smooth manifolds such that f is # a surjective submersion, and # a proper map, (in particular, this condition is always satisfied if M is compact), is a locally trivial fibration. This is a foundational result in differential topology, and exists in many further variants. It is due to Charles Ehresmann.
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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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In algebraic geometry, a universal homeomorphism is a morphism of schemes f: X \to Y such that, for each morphism Y' \to Y, the base change X \times_Y Y' \to Y' is a homeomorphism of topological spaces. A morphism of schemes is a universal homeomorphism if and only if it is integral, radicial and surjective.EGA IV4, 18.12.11. In particular, a morphism of locally of finite type is a universal homeomorphism if and only if it is finite, radicial and surjective.
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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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Cubes occasionally have the surjective property in other fields, such as in for such prime that ,The multiplicative group of is cyclic of order , and if it is not divisible by 3, then cubes define a group automorphism. but not necessarily: see the counterexample with rationals above. Also in only three elements 0, ±1 are perfect cubes, of seven total. −1, 0, and 1 are perfect cubes anywhere and the only elements of a field equal to the own cubes: .
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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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Note that the kernel can be equipped with a B-module structure as follows: since p is surjective, any b \in B has a lift to a x\in E , so b \cdot m := x\cdot m for m \in I. Since any lift differs by an element k \in I in the kernel, and > (x + k)\cdot m = x\cdot m + k\cdot m = x\cdot m because the ideal is square-zero, this module structure is well-defined.
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The group Spin(3) is isomorphic to the special unitary group SU(2); it is also diffeomorphic to the unit 3-sphere S3 and can be understood as the group of unit quaternions (i.e. those with absolute value 1). The connection between quaternions and rotations, commonly exploited in computer graphics, is explained in quaternions and spatial rotations. The map from S3 onto SO(3) that identifies antipodal points of S3 is a surjective homomorphism of Lie groups, with kernel {±1}.
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A constant function is both monotone and antitone; conversely, if f is both monotone and antitone, and if the domain of f is a lattice, then f must be constant. Monotone functions are central in order theory. They appear in most articles on the subject and examples from special applications are found in these places. Some notable special monotone functions are order embeddings (functions for which x ≤ y if and only if f(x) ≤ f(y)) and order isomorphisms (surjective order embeddings).
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They are defined as surjective homomorphisms (i.e., something maps to each vertex) that are also locally bijective, that is, a bijection on the neighbourhood of each vertex. An example is the bipartite double cover, formed from a graph by splitting each vertex v into v0 and v1 and replacing each edge u,v with edges u0,v1 and v0,u1. The function mapping v0 and v1 in the cover to v in the original graph is a homomorphism and a covering map.
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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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A classic result in semigroup theory due to D. B. McAlister states that every inverse semigroup has an E-unitary cover; besides being surjective, the homomorphism in this case is also idempotent separating, meaning that in its kernel an idempotent and non-idempotent never belong to the same equivalence class.; something slightly stronger has actually be shown for inverse semigroups: every inverse semigroup admits an F-inverse cover.Lawson p. 230 McAlister's covering theorem generalizes to orthodox semigroups: every orthodox semigroup has a unitary cover.
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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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A codomain is part of a function if is defined as a triple where is called the domain of , its codomain, and its graph. The set of all elements of the form , where ranges over the elements of the domain , is called the image of . The image of a function is a subset of its codomain so it might not coincide with it. Namely, a function that is not surjective has elements in its codomain for which the equation does not have a solution.
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Proposition 1. So uniruledness does not imply that the Kodaira dimension is −∞ in positive characteristic. A variety X is separably uniruled if there is a variety Y with a dominant separable rational map Y × P1 – → X which does not factor through the projection to Y. ("Separable" means that the derivative is surjective at some point; this would be automatic for a dominant rational map in characteristic zero.) A separably uniruled variety has Kodaira dimension −∞. The converse is true in dimension 2, but not in higher dimensions.
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The Bethe lattice or infinite Cayley tree is the Cayley graph of the free group on n generators. A presentation of a group G by n generators corresponds to a surjective map from the free group on n generators to the group G, and at the level of Cayley graphs to a map from the infinite Cayley tree to the Cayley graph. This can also be interpreted (in algebraic topology) as the universal cover of the Cayley graph, which is not in general simply connected.
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Assume that a group, , is a group extension given as a short exact sequence of groups : with kernel, , and quotient, . If the kernel, , is a complete group then the extension splits: is isomorphic to the direct product, . A proof using homomorphisms and exact sequences can be given in a natural way: The action of (by conjugation) on the normal subgroup, gives rise to a group homomorphism, . Since and has trivial center the homomorphism is surjective and has an obvious section given by the inclusion of in .
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This is equivalent to the above notion, as every dense morphism between two abelian varieties of the same dimension is automatically surjective with finite fibres, and if it preserves identities then it is a homomorphism of groups. Two abelian varieties E1 and E2 are called isogenous if there is an isogeny E1 → E2. This is an equivalence relation, symmetry being due to the existence of the dual isogeny. As above, every isogeny induces homomorphisms of the groups of the k-valued points of the abelian varieties.
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Equivalently, f is a quotient map if and only if it is the transfinite composition of maps X\rightarrow X/Z, where Z\subset X is a subset. Note that this doesn't imply that f is an open function. ;Quotient space: If X is a space, Y is a set, and f : X → Y is any surjective function, then the quotient topology on Y induced by f is the finest topology for which f is continuous. The space X is a quotient space or identification space.
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Every unit quaternion is naturally associated to a spatial rotation in 3 dimensions, and the product of two quaternions is associated to the composition of the associated rotations. Furthermore, every rotation arises from exactly two unit quaternions in this fashion. In short: there is a 2:1 surjective homomorphism from SU(2) to SO(3); consequently SO(3) is isomorphic to the quotient group SU(2)/{±I}, the manifold underlying SO(3) is obtained by identifying antipodal points of the 3-sphere , and SU(2) is the universal cover of SO(3).
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For this reason, it is convenient to define the spin representations over the complex numbers first, and derive real representations by introducing real structures. The properties of the spin representations depend, in a subtle way, on the dimension and signature of the orthogonal group. In particular, spin representations often admit invariant bilinear forms, which can be used to embed the spin groups into classical Lie groups. In low dimensions, these embeddings are surjective and determine special isomorphisms between the spin groups and more familiar Lie groups; this elucidates the properties of spinors in these dimensions.
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Any general property of Banach spaces continues to hold for Hilbert spaces. The open mapping theorem states that a continuous surjective linear transformation from one Banach space to another is an open mapping meaning that it sends open sets to open sets. A corollary is the bounded inverse theorem, that a continuous and bijective linear function from one Banach space to another is an isomorphism (that is, a continuous linear map whose inverse is also continuous). This theorem is considerably simpler to prove in the case of Hilbert spaces than in general Banach spaces.
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Let F be finitely generated free group, with n generators. Let G1 and G2 be two finitely presented groups. Suppose there exists a surjective homomorphism \phi:F\rightarrow G_1\ast G_2, then there exists two subgroups F1 and F2 of F with \phi(F_1)=G_1 and \phi(F_2)=G_2 such that F=F_1\ast F_2. Proof: We give the proof assuming that F has no generator which is mapped to the identity of G_1\ast G_2, for if there are such generators, they may be added to any of F_1 or F_2.
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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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An isogeny from an abelian variety A to another one B is a surjective morphism with finite kernel. Some theorems on abelian varieties require the idea of abelian variety up to isogeny for their convenient statement. For example, given an abelian subvariety A1 of A, there is another subvariety A2 of A such that :A1 × A2 is isogenous to A (Poincaré's reducibility theorem: see for example Abelian Varieties by David Mumford). To call this a direct sum decomposition, we should work in the category of abelian varieties up to isogeny.
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However, it is often useful to see functions as mappings, which consist not only of the relation between input and output, but also which set is the domain, and which set is the codomain. For example, to say that a function is onto (surjective) or not the codomain should be taken into account. The graph of a function on its own doesn't determine the codomain. It is common to use both terms function and graph of a function since even if considered the same object, they indicate viewing it from a different perspective.
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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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An R-module P is projective if and only if the covariant functor is an exact functor, where R-Mod is the category of left R-modules and Ab is the category of abelian groups. When the ring R is commutative, Ab is advantageously replaced by R-Mod in the preceding characterization. This functor is always left exact, but, when P is projective, it is also right exact. This means that P is projective if and only if this functor preserves epimorphisms (surjective homomorphisms), or if it preserves finite colimits.
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Continue this process whereby choosing a neighbourhood Un+1 ⊂ Un whose closure does not contain xn+1. Then the collection {Ui : i ∈ N} satisfies the finite intersection property and hence the intersection of their closures is non-empty by the compactness of X. Therefore, there is a point x in this intersection. No xi can belong to this intersection because xi does not belong to the closure of Ui. This means that x is not equal to xi for all i and f is not surjective; a contradiction. Therefore, X is uncountable.
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A subdirect product is a subalgebra (in the sense of universal algebra) A of a direct product ΠiAi such that every induced projection (the composite pjs: A → Aj of a projection pj: ΠiAi → Aj with the subalgebra inclusion s: A → ΠiAi) is surjective. A direct (subdirect) representation of an algebra A is a direct (subdirect) product isomorphic to A. An algebra is called subdirectly irreducible if it is not subdirectly representable by "simpler" algebras. Subdirect irreducibles are to subdirect product of algebras roughly as primes are to multiplication of integers.
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Let A be a linear operator defined on a linear subspace D(A) of the reflexive Banach space X. Then A generates a contraction semigroup if and only ifEngel and Nagel Corollary II.3.20 # A is dissipative, and # A − λ0I is surjective for some λ0> 0, where I denotes the identity operator. Note that the conditions that D(A) is dense and that A is closed are dropped in comparison to the non-reflexive case. This is because in the reflexive case they follow from the other two conditions.
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The "upward" property of filters is less important for topologicial intuition but it is sometimes useful to have for technical reasons. ;Prefilters vs. filters With respect to maps and subsets, the property of being a prefilter is in general more well behaved and better preserved than the property of being a filter. For instance, the image of a prefilter under some map is again a prefilter; but the image of a filter under a non-surjective map is never a filter on the codomain, although it will be a prefilter.
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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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Note that the half-period ratio can be thought of as a simple number, namely, one of the parameters to elliptic functions, or it can be thought of as a function itself, because the half periods can be given in terms of the elliptic modulus or in terms of the nome. This follows because Klein's j-invariant is surjective onto the complex plane; it gives a bijection between isomorphism classes of elliptic curves and the complex numbers. See the pages on quarter period and elliptic integrals for additional definitions and relations on the arguments and parameters to elliptic functions.
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For instance, the completion of a metric space M involves an isometry from M into M', a quotient set of the space of Cauchy sequences on M. The original space M is thus isometrically isomorphic to a subspace of a complete metric space, and it is usually identified with this subspace. Other embedding constructions show that every metric space is isometrically isomorphic to a closed subset of some normed vector space and that every complete metric space is isometrically isomorphic to a closed subset of some Banach space. An isometric surjective linear operator on a Hilbert space is called a unitary operator.
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Let π:E→M be a smooth fiber bundle over a smooth manifold M. The vertical bundle is the kernel VE := ker(dπ) of the tangent map dπ : TE → TM. (page 77) Since dπe is surjective at each point e, it yields a regular subbundle of TE. Furthermore, the vertical bundle VE is also integrable. An Ehresmann connection on E is a choice of a complementary subbundle HE to VE in TE, called the horizontal bundle of the connection. At each point e in E, the two subspaces form a direct sum, such that TeE = VeE ⊕ HeE.
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Let πE:E→ M and πF:F→ N be fiber bundles over spaces M and N respectively. Then a continuous map \varphi : E \to F is called a bundle map from E to F if there is a continuous map f:M→ N such that the diagram center commutes, that is, \pi_F\circ\varphi = f\circ\pi_E . In other words, \varphi is fiber-preserving, and f is the induced map on the space of fibers of E: since πE is surjective, f is uniquely determined by \varphi. For a given f, such a bundle map \varphi is said to be a bundle map covering f.
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Consequently, to define an étale cover of a scheme X, it suffices to first cover X by open affine subschemes, that is, to take a Zariski cover, and then to define an étale cover of an affine scheme. An étale cover of an affine scheme X can be defined as a surjective family {uα : Xα → X} such that the set of all α is finite, each Xα is affine, and each uα is étale. Then an étale cover of X is a family {uα : Xα → X} which becomes an étale cover after base changing to any open affine subscheme of X.
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The Bethe lattice where each node is joined to 2n others is essentially the Cayley graph of a free group on n generators. It is an infinite Cayley tree. A presentation of a group G by n generators corresponds to a surjective map from the free group on n generators to the group G, and at the level of Cayley graphs to a map from the Bethe lattice (with distinguished root corresponding to the identity) to the Cayley graph. This can also be interpreted (in algebraic topology) as the universal cover of the Cayley graph, which is not in general simply connected.
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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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A curve , over is called a modular curve if for some there exists a surjective morphism , given by a rational map with integer coefficients. The famous modularity theorem tells us that all elliptic curves over are modular. Mappings also arise in connection with since points on it correspond to some -isogenous pairs of elliptic curves. An isogeny between two elliptic curves is a non-trivial morphism of varieties (defined by a rational map) between the curves which also respects the group laws, and hence which sends the point at infinity (serving as the identity of the group law) to the point at infinity.
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An elliptic surface is a surface equipped with an elliptic fibration (a surjective holomorphic map to a curve B such that all but finitely many fibers are smooth irreducible curves of genus 1). The generic fiber in such a fibration is a genus 1 curve over the function field of B. Conversely, given a genus 1 curve over the function field of a curve, its relative minimal model is an elliptic surface. Kodaira and others have given a fairly complete description of all elliptic surfaces. In particular, Kodaira gave a complete list of the possible singular fibers.
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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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The canonical map is the quotient map , which is continuous when has either the norm topology or the quotient topology. If and are radial disks such that then ⊆ p(0) so there is a continuous linear surjective canonical map defined by sending to the equivalence class , where one may verify that the definition does not depend on the representative of the equivalence class that is chosen. This canonical map has norm and it has a unique continuous linear canonical extension to that is denoted by . Suppose that in addition and are bounded disks in with so that and the natural inclusion is a continuous linear map.
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It is also possible to define an order isomorphism to be a surjective order-embedding. The two assumptions that f cover all the elements of T and that it preserve orderings, are enough to ensure that f is also one-to-one, for if f(x)=f(y) then (by the assumption that f preserves the order) it would follow that x\le y and y\le x, implying by the definition of a partial order that x=y. Yet another characterization of order isomorphisms is that they are exactly the monotone bijections that have a monotone inverse.This is the definition used by and .
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Every topological space is a dense subset of itself. For a set X equipped with the discrete topology, the whole space is the only dense subset. Every non- empty subset of a set X equipped with the trivial topology is dense, and every topology for which every non-empty subset is dense must be trivial. Denseness is transitive: Given three subsets A, B and C of a topological space X with such that A is dense in B and B is dense in C (in the respective subspace topology) then A is also dense in C. The image of a dense subset under a surjective continuous function is again dense.
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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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For instance, Fulton, in his book on intersection theory, uses the definition above. In literature, however, a correspondence from a variety X to a variety Y is often taken to be a subset Z of X×Y such that Z is finite and surjective over each component of X. Note the asymmetry in this latter definition; which talks about a correspondence from X to Y rather than a correspondence between X and Y. The typical example of the latter kind of correspondence is the graph of a function f:X→Y. Correspondences also play an important role in the construction of motives (cf. presheaf with transfers).
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The condition Ext1(A, Z) = 0 can be equivalently formulated as follows: whenever B is an abelian group and f : B → A is a surjective group homomorphism whose kernel is isomorphic to the group of integers Z, then there exists a group homomorphism g : A → B with fg = idA. Abelian groups A satisfying this condition are sometimes called Whitehead groups, so Whitehead's problem asks: is every Whitehead group free? Caution: The converse of Whitehead's problem, namely that every free abelian group is Whitehead, is a well known group-theoretical fact. Some authors call Whitehead group only a non-free group A satisfying Ext1(A, Z) = 0.
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In mathematical set theory, an ω-Jónsson function for a set x of ordinals is a function f:[x]^\omega\to x with the property that, for any subset y of x with the same cardinality as x, the restriction of f to [y]^\omega is surjective on x. Here [x]^\omega denotes the set of strictly increasing sequences of members of x, or equivalently the family of subsets of x with order type \omega, using a standard notation for the family of subsets with a given order type. Jónsson functions are named for Bjarni Jónsson. showed that for every ordinal λ there is an ω-Jónsson function for λ.
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That is, F is an étale sheaf if and only if the following condition is true. Suppose that is an object of Ét(X) and that is a jointly surjective family of étale morphisms over X. For each i, choose a section xi of F over Ui. The projection map , which is loosely speaking the inclusion of the intersection of Ui and Uj in Ui, induces a restriction map . If for all i and j the restrictions of xi and xj to are equal, then there must exist a unique section x of F over U which restricts to xi for all i. Suppose that X is a Noetherian scheme.
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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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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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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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Suppose that G is a profinite group acting on a module A with a surjective homomorphism π from the G-module A to itself. Suppose also that G acts trivially on the kernel C of π and that the first cohomology group H1(G,A) is trivial. Then the exact sequence of group cohomology shows that there is an isomorphism between AG/π(AG) and Hom(G,C). Kummer theory is the special case of this when A is the multiplicative group of the separable closure of a field k, G is the Galois group, π is the nth power map, and C the group of nth roots of unity.
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This is surjective, and a local isomorphism, but not an isomorphism (in particular because D is Hausdorff and N is not). However, the global Torelli theorem for K3 surfaces says that the quotient map of sets :N/O(\Lambda)\to D/O(\Lambda) is bijective. It follows that two complex analytic K3 surfaces X and Y are isomorphic if and only if there is a Hodge isometry from H^2(X,\Z) to H^2(Y,\Z), that is, an isomorphism of abelian groups that preserves the intersection form and sends H^0(X,\Omega^2)\subset H^2(X,\Complex) to H^0(Y,\Omega^2).
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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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In the context of schemes, the importance of ideal sheaves lies mainly in the correspondence between closed subschemes and quasi-coherent ideal sheaves. Consider a scheme X and a quasi-coherent ideal sheaf J in OX. Then, the support Z of OX/J is a closed subspace of X, and (Z, OX/J) is a scheme (both assertions can be checked locally). It is called the closed subscheme of X defined by J. Conversely, let i: Z → X be a closed immersion, i.e., a morphism which is a homeomorphism onto a closed subspace such that the associated map : i#: OX → i⋆OZ is surjective on the stalks.
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In mathematics, perfectoid spaces are adic spaces of special kind, which occur in the study of problems of "mixed characteristic", such as local fields of characteristic zero which have residue fields of characteristic prime p. A perfectoid field is a complete topological field K whose topology is induced by a nondiscrete valuation of rank 1, such that the Frobenius endomorphism Φ is surjective on K°/p where K° denotes the ring of power-bounded elements. Perfectoid spaces may be used to (and were invented in order to) compare mixed characteristic situations with purely finite characteristic ones. Technical tools for making this precise are the tilting equivalence and the almost purity theorem.
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SU(2) is the universal covering group of SO(3), and so its representation theory includes that of the latter, by dint of a surjective homomorphism to it. This underlies the significance of SU(2) for the description of non-relativistic spin in theoretical physics; see below for other physical and historical context. As shown below, the finite-dimensional irreducible representations of SU(2) are indexed by a non-negative integer m and have dimension m+1. In the physics literature, the representations are labeled by the quantity l=m/2, where l is then either an integer or a half- integer, and the dimension is 2l+1.
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Diagram of the fundamental theorem on homomorphisms where f is a homomorphism, N is a normal subgroup of G and e is the identity element of G Given two groups G and H and a group homomorphism f : G→H, let K be a normal subgroup in G and φ the natural surjective homomorphism G→G/K (where G/K is a quotient group). If K is a subset of ker(f) then there exists a unique homomorphism h:G/K→H such that f = h φ. In other words, the natural projection φ is universal among homomorphisms on G that map K to the identity element. The situation is described by the following commutative diagram: File:Fundamental Homomorphism Theorem.
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A local Noetherian ring is regular if and only if its global dimension is finite, say n, which means that any finitely generated R-module has a resolution by projective modules of length at most n. The proof of this and other related statements relies on the usage of homological methods, such as the Ext functor. This functor is the derived functor of the functor :HomR(M, −). The latter functor is exact if M is projective, but not otherwise: for a surjective map E -> F of R-modules, a map M -> F need not extend to a map M -> E. The higher Ext functors measure the non-exactness of the Hom-functor.
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In this language, the definition of the étale topology is succinct but abstract: It is the topology generated by the pretopology whose covering families are jointly surjective families of étale morphisms. The small étale site of X is the category O(Xét) whose objects are schemes U with a fixed étale morphism U → X. The morphisms are morphisms of schemes compatible with the fixed maps to X. The big étale site of X is the category Ét/X, that is, the category of schemes with a fixed map to X, considered with the étale topology. The étale topology can be defined using slightly less data. First, notice that the étale topology is finer than the Zariski topology.
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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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As noted already, when is less than , , the trivial group . The reason is that a continuous mapping from an -sphere to an -sphere with can always be deformed so that it is not surjective. Consequently, its image is contained in with a point removed; this is a contractible space, and any mapping to such a space can be deformed into a one-point mapping. The case has also been noted already, and is an easy consequence of the Hurewicz theorem: this theorem links homotopy groups with homology groups, which are generally easier to calculate; in particular, it shows that for a simply-connected space X, the first nonzero homotopy group , with , is isomorphic to the first nonzero homology group .
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A densely defined operator T is symmetric, if the subspace (defined in a previous section) is orthogonal to its image under J (where J(x,y):=(y,-x)).Follows from and the definition via adjoint operators. Equivalently, an operator T is self-adjoint if it is densely defined, closed, symmetric, and satisfies the fourth condition: both operators , are surjective, that is, map the domain of T onto the whole space H. In other words: for every x in H there exist y and z in the domain of T such that and . An operator T is self-adjoint, if the two subspaces , are orthogonal and their sum is the whole space H \oplus H .
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The maps between the kernels and the maps between the cokernels are induced in a natural manner by the given (horizontal) maps because of the diagram's commutativity. The exactness of the two induced sequences follows in a straightforward way from the exactness of the rows of the original diagram. The important statement of the lemma is that a connecting homomorphism d exists which completes the exact sequence. In the case of abelian groups or modules over some ring, the map d can be constructed as follows: Pick an element x in ker c and view it as an element of C; since g is surjective, there exists y in B with g(y) = x.
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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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This implies that their fundamental groups are trivial, and all homology groups are trivial except the 0th one, which is isomorphic to Z. The Euler characteristic of a point (and therefore also that of a closed or open disk) is 1.In higher dimensions, the Euler characteristic of a closed ball remains equal to +1, but the Euler characteristic of an open ball is +1 for even-dimensional balls and −1 for odd-dimensional balls. See . Every continuous map from the closed disk to itself has at least one fixed point (we don't require the map to be bijective or even surjective); this is the case n=2 of the Brouwer fixed point theorem.
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In order to produce a computable real, a Turing machine must compute a total function, but the corresponding decision problem is in Turing degree 0′′. Consequently, there is no surjective computable function from the natural numbers to the computable reals, and Cantor's diagonal argument cannot be used constructively to demonstrate uncountably many of them. While the set of real numbers is uncountable, the set of computable numbers is classically countable and thus almost all real numbers are not computable. Here, for any given computable number x, the well ordering principle provides that there is a minimal element in S which corresponds to x, and therefore there exists a subset consisting of the minimal elements, on which the map is a bijection.
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Another construction of ΛR takes somewhat longer to describe, but better indicates the relationship with the rings R[X1,...,Xn]Sn of symmetric polynomials in n indeterminates. For every n there is a surjective ring homomorphism ρn from the analogous ring R[X1,...,Xn+1]Sn+1 with one more indeterminate onto R[X1,...,Xn]Sn, defined by setting the last indeterminate Xn+1 to 0\. Although ρn has a non-trivial kernel, the nonzero elements of that kernel have degree at least n+1 (they are multiples of X1X2...Xn+1). This means that the restriction of ρn to elements of degree at most n is a bijective linear map, and ρn(ek(X1,...,Xn+1)) = ek(X1,...,Xn) for all k ≤ n.
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Quotient maps are characterized among surjective maps by the following property: if Z is any topological space and is any function, then f is continuous if and only if is continuous. Characteristic property of the quotient topology The quotient space X/~ together with the quotient map is characterized by the following universal property: if is a continuous map such that implies for all a and b in X, then there exists a unique continuous map such that . We say that g descends to the quotient. The continuous maps defined on X/~ are therefore precisely those maps which arise from continuous maps defined on X that respect the equivalence relation (in the sense that they send equivalent elements to the same image).
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A linear algebraic group G over a field k is called simple (or k-simple) if it is semisimple, nontrivial, and every smooth connected normal subgroup of G over k is trivial or equal to G.Conrad (2014), after Proposition 5.1.17. (Some authors call this property "almost simple".) This differs slightly from the terminology for abstract groups, in that a simple algebraic group may have nontrivial center (although the center must be finite). For example, for any integer n at least 2 and any field k, the group SL(n) over k is simple, and its center is the group scheme μn of nth roots of unity. A central isogeny of reductive groups is a surjective homomorphism with kernel a finite central subgroup scheme.
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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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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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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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An RA is a Q-relation algebra (QRA) if, in addition to B1-B10, there exist some A and B such that (Tarski and Givant 1987: §8.4): :Q0: A˘•A ≤ I :Q1: B˘•B ≤ I :Q2: A˘•B = 1 Essentially these axioms imply that the universe has a (non- surjective) pairing relation whose projections are A and B. It is a theorem that every QRA is a RRA (Proof by Maddux, see Tarski & Givant 1987: 8.4(iii) ). Every QRA is representable (Tarski and Givant 1987). That not every relation algebra is representable is a fundamental way RA differs from QRA and Boolean algebras, which, by Stone's representation theorem for Boolean algebras, are always representable as sets of subsets of some set, closed under union, intersection, and complement.
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For an orientable surface S the Euler characteristic χ(S) is :2-2g \, where g is the genus (the number of handles), since the Betti numbers are 1, 2g, 1, 0, 0, ... . In the case of an (unramified) covering map of surfaces :\pi:S' \to S \, that is surjective and of degree N, we should have the formula :\chi(S') = N\cdot\chi(S). \, That is because each simplex of S should be covered by exactly N in S′ -- at least if we use a fine enough triangulation of S, as we are entitled to do since the Euler characteristic is a topological invariant. What the Riemann–Hurwitz formula does is to add in a correction to allow for ramification (sheets coming together).
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For a compact, connected, orientable surface S, the Euler characteristic \chi(S) is :\chi(S)=2-2g, where g is the genus (the number of handles), since the Betti numbers are 1, 2g, 1, 0, 0, \dots. In the case of an (unramified) covering map of surfaces :\pi\colon S' \to S that is surjective and of degree N, we have the formula :\chi(S') = N\cdot\chi(S). That is because each simplex of S should be covered by exactly N in S', at least if we use a fine enough triangulation of S, as we are entitled to do since the Euler characteristic is a topological invariant. What the Riemann–Hurwitz formula does is to add in a correction to allow for ramification (sheets coming together).
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Proceeding from the one necessarily true and unquestionable fact – that we are present to our experiences – an understanding of reality is developed that is neither a materialist nor an idealist conceptualization. This way of viewing the world is referred to as surjective, a metaphorical use of a concept found in mathematical set theory that means a function that works upon every member of a set, where Awareness is the function and Omnific Awareness is the set, in order to distinguish this position from both subjectivity and objectivity. Within this system anything whatsoever can arise from Omnific Awareness, thus the use of the term “indefinite” in labeling this monism. What does arise as the existents that we are conscious of is conditioned by the affections of Awareness for its display.
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Then M does not admit a Kähler structure if and only if M admits a non-zero positive (1,1)-current \Theta which is a (1,1)-part of an exact 2-current. Note that the de Rham differential maps 3-currents to 2-currents, hence \Theta is a differential of a 3-current; if \Theta is a current of integration of a complex curve, this means that this curve is a (1,1)-part of a boundary. When M admits a surjective map \pi:\; M \mapsto X to a Kähler manifold with 1-dimensional fibers, this theorem leads to the following result of complex algebraic geometry. Corollary: In this situation, M is non-Kähler if and only if the homology class of a generic fiber of \pi is a (1,1)-part of a boundary.
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If we write F× for the multiplicative group of F (excluding 0), then the determinant is a group homomorphism :det: GL(n, F) → F×. that is surjective and its kernel is the special linear group. Therefore, by the first isomorphism theorem, is isomorphic to F×. In fact, can be written as a semidirect product: :GL(n, F) = SL(n, F) ⋊ F× The special linear group is also the derived group (also known as commutator subgroup) of the GL(n, F) (for a field or a division ring F) provided that n e 2 or k is not the field with two elements., Theorem II.9.4 When F is R or C, is a Lie subgroup of of dimension . The Lie algebra of consists of all matrices over F with vanishing trace.
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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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The fiber of the corresponding vector bundle over x is then the range of f(x). If M is not connected, the converse does not hold unless one allows for vector bundles of non-constant rank (which means admitting manifolds of non-constant dimension). For example, if M is a zero-dimensional 2-point manifold, the module \R\oplus 0 is finitely-generated and projective over C^\infty(M)\cong\R\times\R but is not free, and so cannot correspond to the sections of any (constant-rank) vector bundle over M (all of which are trivial). Another way of stating the above is that for any connected smooth manifold M, the section functor Γ from the category of smooth vector bundles over M to the category of finitely generated, projective C∞(M)-modules is full, faithful, and essentially surjective.
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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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The mapping cylinder may be viewed as a way to replace an arbitrary map by an equivalent cofibration, in the following sense: Given a map f\colon X \to Y, the mapping cylinder is a space M_f, together with a cofibration \tilde f\colon X \to M_f and a surjective homotopy equivalence M_f \to Y (indeed, Y is a deformation retract of M_f), such that the composition X \to M_f \to Y equals f. right Thus the space Y gets replaced with a homotopy equivalent space M_f, and the map f with a lifted map \tilde f. Equivalently, the diagram :f\colon X \to Y gets replaced with a diagram :\tilde f\colon X \to M_f together with a homotopy equivalence between them. The construction serves to replace any map of topological spaces by a homotopy equivalent cofibration.
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The quotient group is isomorphic to the trivial group (the group with one element), and is isomorphic to G. The order of , by definition the number of elements, is equal to , the index of N in G. If G is finite, the index is also equal to the order of G divided by the order of N. The set may be finite, although both G and N are infinite (for example, ). There is a "natural" surjective group homomorphism , sending each element g of G to the coset of N to which g belongs, that is: . The mapping π is sometimes called the canonical projection of G onto . Its kernel is N. There is a bijective correspondence between the subgroups of G that contain N and the subgroups of ; if H is a subgroup of G containing N, then the corresponding subgroup of is π(H).
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Let f\colon A \to B be a homomorphism. We want to prove that if it is not surjective, it is not right cancelable. In the case of sets, let b be an element of B that not belongs to f(A), and define g, h\colon B \to B such that g is the identity function, and that h(x) = x for every x \in B, except that h(b) is any other element of B. Clearly f is not right cancelable, as g eq h and g \circ f = h \circ f. In the case of vector spaces, abelian groups and modules, the proof relies on the existence of cokernels and on the fact that the zero maps are homomorphisms: let C be the cokernel of f, and g\colon B \to C be the canonical map, such that g(f(A)) = 0.
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More precisely, let V be an algebraic variety over K (assumptions here are: V is an irreducible set, a quasi-projective variety, and K has characteristic zero). A type I thin set is a subset of V(K) that is not Zariski-dense. That means it lies in an algebraic set that is a finite union of algebraic varieties of dimension lower than d, the dimension of V. A type II thin set is an image of an algebraic morphism (essentially a polynomial mapping) φ, applied to the K-points of some other d-dimensional algebraic variety V′, that maps essentially onto V as a ramified covering with degree e > 1. Saying this more technically, a thin set of type II is any subset of :φ(V′(K)) where V′ satisfies the same assumptions as V and φ is generically surjective from the geometer's point of view.
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In the area of mathematics known as functional analysis, a reflexive space is a locally convex topological vector space (TVS) such that the canonical evaluation map from X into its bidual (which is the strong dual of the strong dual of X) is an isomorphism of TVSs. Since a normable TVS is reflexive if and only if it is semi-reflexive, every normed space (and so in particular, every Banach space) X is reflexive if and only if the canonical evaluation map from X into its bidual is surjective; in this case the normed space is necessarily also a Banach space. Note that in 1951, R. C. James discovered a non-reflexive Banach space that is isometrically isomorphic to its bidual (any such isomorphism is thus necessarily not the canonical evaluation map). Reflexive spaces play an important role in the general theory of locally convex TVSs and in the theory of Banach spaces in particular.
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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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In general, a graph may have multiple double covers that are different from the bipartite double cover.. In the following figure, the graph C is a double cover of the graph H: # The graph C is a covering graph of H: there is a surjective local isomorphism f from C to H, the one indicated by the colours. For example, f maps both blue nodes in C to the blue node in H. Furthermore, let X be the neighbourhood of a blue node in C and let Y be the neighbourhood of the blue node in H; then the restriction of f to X is a bijection from X to Y. In particular, the degree of each blue node is the same. The same applies to each colour. # The graph C is a double cover (or 2-fold cover or 2-lift) of H: the preimage of each node in H has size 2.
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Some important properties P of morphisms of schemes are preserved under arbitrary base change. That is, if X → Y has property P and Z → Y is any morphism of schemes, then the base change X xY Z → Z has property P. For example, flat morphisms, smooth morphisms, proper morphisms, and many other classes of morphisms are preserved under arbitrary base change.. The word descent refers to the reverse question: if the pulled-back morphism X xY Z → Z has some property P, must the original morphism X → Y have property P? Clearly this is impossible in general: for example, Z might be the empty scheme, in which case the pulled- back morphism loses all information about the original morphism. But if the morphism Z → Y is flat and surjective (also called faithfully flat) and quasi- compact, then many properties do descend from Z to Y. Properties that descend include flatness, smoothness, properness, and many other classes of morphisms.. These results form part of Grothendieck's theory of faithfully flat descent.
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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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If is a product of rings, then for every i in I we have a surjective ring homomorphism which projects the product on the ith coordinate. The product R, together with the projections pi, has the following universal property: :if S is any ring and is a ring homomorphism for every i in I, then there exists precisely one ring homomorphism such that for every i in I. This shows that the product of rings is an instance of products in the sense of category theory. When I is finite, the underlying additive group of coincides with the direct sum of the additive groups of the Ri. In this case, some authors call R the "direct sum of the rings Ri" and write , but this is incorrect from the point of view of category theory, since it is usually not a coproduct in the category of rings: for example, when two or more of the Ri are nonzero, the inclusion map fails to map 1 to 1 and hence is not a ring homomorphism. (A finite coproduct in the category of commutative (associative) algebras over a commutative ring is a tensor product of algebras.
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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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A quotient space is defined as follows: if X is a topological space and Y is a set, and if f : X→ Y is a surjective function, then the quotient topology on Y is the collection of subsets of Y that have open inverse images under f. In other words, the quotient topology is the finest topology on Y for which f is continuous. A common example of a quotient topology is when an equivalence relation is defined on the topological space X. The map f is then the natural projection onto the set of equivalence classes. The Vietoris topology on the set of all non-empty subsets of a topological space X, named for Leopold Vietoris, is generated by the following basis: for every n-tuple U1, ..., Un of open sets in X, we construct a basis set consisting of all subsets of the union of the Ui that have non-empty intersections with each Ui. The Fell topology on the set of all non-empty closed subsets of a locally compact Polish space X is a variant of the Vietoris topology, and is named after mathematician James Fell.
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E(2) is a semidirect product of O(2) and the translation group T. In other words, O(2) is a subgroup of E(2) isomorphic to the quotient group of E(2) by T: :O(2) \cong E(2) / T There is a "natural" surjective group homomorphism p : E(2) → E(2)/ T, sending each element g of E(2) to the coset of T to which g belongs, that is: p (g) = gT, sometimes called the canonical projection of E(2) onto E(2) / T or O(2). Its kernel is T. For every subgroup of E(2) we can consider its image under p: a point group consisting of the cosets to which the elements of the subgroup belong, in other words, the point group obtained by ignoring translational parts of isometries. For every discrete subgroup of E(2), due to the crystallographic restriction theorem, this point group is either Cn or of type Dn for n = 1, 2, 3, 4, or 6. Cn and Dn for n = 1, 2, 3, 4, and 6 can be combined with translational symmetry, sometimes in more than one way. Thus these 10 groups give rise to 17 wallpaper groups, and the four groups with n = 1 and 2, give also rise to 7 frieze groups.
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