A generalization defines an order on a Cartesian product of partially ordered sets; this order is a total order if and only if all factors of the Cartesian product are totally ordered.
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One can similarly define the Cartesian product of n sets, also known as an n-fold Cartesian product, which can be represented by an n-dimensional array, where each element is an n-tuple. An ordered pair is a 2-tuple or couple. More generally still, one can define the Cartesian product of an indexed family of sets. The Cartesian product is named after René Descartes, whose formulation of analytic geometry gave rise to the concept, which is further generalized in terms of direct product.
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Although the Cartesian product is traditionally applied to sets, category theory provides a more general interpretation of the product of mathematical structures. This is distinct from, although related to, the notion of a Cartesian square in category theory, which is a generalization of the fiber product. Exponentiation is the right adjoint of the Cartesian product; thus any category with a Cartesian product (and a final object) is a Cartesian closed category.
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The graph of a function is contained in a Cartesian product of sets. An X–Y plane is a cartesian product of two lines, called X and Y, while a cylinder is a cartesian product of a line and a circle, whose height, radius, and angle assign precise locations of the points. Fibre bundles are not Cartesian products, but appear to be up close. There is a corresponding notion of a graph on a fibre bundle called a section.
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In graph theory, the Cartesian product of two graphs G and H is the graph denoted by , whose vertex set is the (ordinary) Cartesian product and such that two vertices (u,v) and (u′,v′) are adjacent in , if and only if and v is adjacent with v′ in H, or and u is adjacent with u′ in G. The Cartesian product of graphs is not a product in the sense of category theory. Instead, the categorical product is known as the tensor product of graphs.
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In geometry of 6 dimensions or higher, a triple product is a polytope resulting from the Cartesian product of three polytopes, each of two dimensions or higher. The Cartesian product of an a-polytope, a b-polytope, and a c-polytope is an (a + b + c)-polytope, where a, b and c are 2-polytopes (polygon) or higher. The lowest-dimensional forms are 6-polytopes being the Cartesian product of three polygons. The smallest can be written as {3} × {3} × {3} in Schläfli symbols if they are regular, and contains 27 vertices.
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The lexicographical order defines an order on a Cartesian product of ordered sets, which is a total order when all these sets are themselves totally ordered. An element of a Cartesian product is a sequence whose th element belongs to for every . As evaluating the lexicographical order of sequences compares only elements which have the same rank in the sequences, the lexicographical order extends to Cartesian products of ordered sets. Specifically, given two partially ordered sets and , the lexicographical order on the Cartesian product is defined as : if and only if or .
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Just as the binary Cartesian product is readily generalized to an n-ary Cartesian product, binary product of two categories can be generalized, completely analogously, to a product of n categories. The product operation on categories is commutative and associative, up to isomorphism, and so this generalization brings nothing new from a theoretical point of view.
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In mathematics, a Liouville space, also known as a line space or an extended Hilbert space is Cartesian product of two Hilbert spaces.
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Viewing a graph as a category whose objects are the vertices and whose morphisms are the paths in the graph, the cartesian product of graphs corresponds to the funny tensor product of categories. The cartesian product of graphs is one of two graph products that turn the category of graphs and graph homomorphisms into a symmetric closed monoidal category (as opposed to merely symmetric monoidal), the other being the tensor product of graphs. The internal hom [G, H] for the cartesian product of graphs has graph homomorphisms from G to H as vertices and "unnatural transformations" between them as edges.
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If f is a function from A to B and g is a function from X to Y, then their Cartesian product is a function from to with : (f\times g)(a, x) = (f(a), g(x)). This can be extended to tuples and infinite collections of functions. This is different from the standard Cartesian product of functions considered as sets.
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We choose . This operation on morphisms is called cartesian product of morphisms. Second, consider the general product functor. For families we should find a morphism .
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Taking history preserving maps as morphisms in the category of prefix orders leads to a notion of product that is not the Cartesian product of the two orders since the Cartesian product is not always a prefix order. Instead, it leads to an arbitrary interleaving of the original prefix orders. The union of two prefix orders is the disjoint union, as it is with partial orders.
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In set theory, a Cartesian product is a mathematical operation which returns a set (or product set) from multiple sets. That is, for sets A and B, the Cartesian product is the set of all ordered pairs —where and . The class of all things (of a given type) that have Cartesian products is called a Cartesian category. Many of these are Cartesian closed categories.
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If M is an m-manifold and N is an n-manifold, the Cartesian product M×N is a (m+n)-manifold when given the product topology.
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Then the resulting graph remains locally linear. The Cartesian product of any two locally linear graphs remains locally linear, because any triangles in the product come from triangles in one or the other factors. For instance, the nine-vertex Paley graph (the graph of the 3-3 duoprism) is the Cartesian product of two triangles. The Hamming graphs H(d,3) are products of d triangles, and again are locally linear.
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If an infinite set is partitioned into finitely many subsets, then at least one of them must be infinite. Any set which can be mapped onto an infinite set is infinite. The Cartesian product of an infinite set and a nonempty set is infinite. The Cartesian product of an infinite number of sets, each containing at least two elements, is either empty or infinite; if the axiom of choice holds, then it is infinite.
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Since Cartesian coordinates are unique and non-ambiguous, the points of a Cartesian plane can be identified with pairs of real numbers; that is with the Cartesian product \R^2 = \R\times\R, where \R is the set of all real numbers. In the same way, the points in any Euclidean space of dimension n be identified with the tuples (lists) of n real numbers, that is, with the Cartesian product \R^n.
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In geometry, a Hanner polytope is a convex polytope constructed recursively by Cartesian product and polar dual operations. Hanner polytopes are named after Olof Hanner, who introduced them in 1956..
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A metric on a Cartesian product of metric spaces, and a norm on a direct product of normed vector spaces, can be defined in various ways, see for example p-norm.
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In mathematics, the Segre embedding is used in projective geometry to consider the cartesian product (of sets) of two projective spaces as a projective variety. It is named after Corrado Segre.
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Named after (1596–1650), French philosopher, mathematician, and scientist, whose formulation of analytic geometry gave rise to the concept of Cartesian product, which was later generalized to the notion of categorical product.
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Proceed by induction in similar fashion proves the claim. For example, J3 is the Cartesian product of simple algebras :J_3 \simeq J_2 \times J_3 / J_2 \simeq J_1 \times J_2/J_1 \times J_3 / J_2.
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If a general point process is defined on some mathematical space and the random marks are defined on another mathematical space, then the marked point process is defined on the Cartesian product of these two spaces. For a marked Poisson point process with independent and identically distributed marks, the marking theorem states that this marked point process is also a (non-marked) Poisson point process defined on the aforementioned Cartesian product of the two mathematical spaces, which is not true for general point processes.
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While the box topology has a somewhat more intuitive definition than the product topology, it satisfies fewer desirable properties. In particular, if all the component spaces are compact, the box topology on their Cartesian product will not necessarily be compact, although the product topology on their Cartesian product will always be compact. In general, the box topology is finer than the product topology, although the two agree in the case of finite direct products (or when all but finitely many of the factors are trivial).
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In a Cartesian product of uncountably many compact Hausdorff spaces with more than one point, a point is never a Baire set, in spite of the fact that it is closed, and therefore a Borel set.
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A similar equality for the cartesian product of graphs was proven by and rediscovered several times afterwards. An exact formula is also known for the lexicographic product of graphs. introduced two stronger conjectures involving unique colorability.
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Real-world objects that approximate a solid torus include O-rings, non- inflatable lifebuoys, ring doughnuts, and bagels. In topology, a ring torus is homeomorphic to the Cartesian product of two circles: S1 × S1, and the latter is taken to be the definition in that context. It is a compact 2-manifold of genus 1. The ring torus is one way to embed this space into Euclidean space, but another way to do this is the Cartesian product of the embedding of S1 in the plane with itself.
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In the mathematical field of category theory, the product of two categories C and D, denoted and called a product category, is an extension of the concept of the Cartesian product of two sets. Product categories are used to define bifunctors and multifunctors.
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In 6 dimensions and above, there are an infinite amount of non-Wythoffian convex uniform polytopes as the Cartesian product of the Grand antiprism in 4 dimensions and a regular polygon in 2 dimensions. It is not yet proven whether or not there are more.
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In mathematics, it is possible to combine several rings into one large product ring. This is done by giving the Cartesian product of a (possibly infinite) family of rings coordinatewise addition and multiplication. The resulting ring is called a direct product of the original rings.
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In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more obvious, topology called the box topology, which can also be given to a product space and which agrees with the product topology when the product is over only finitely many spaces. However, the product topology is "correct" in that it makes the product space a categorical product of its factors, whereas the box topology is too fine; in that sense the product topology is the natural topology on the Cartesian product.
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In mathematics, an idempotent binary relation is a binary relation R on a set X (a subset of Cartesian product X × X) for which the composition of relations R ∘ R is the same as R. Here:p.3 This notion generalizes that of an idempotent function to relations.
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It is possible to explore versions of kappa calculus with substructural types such as linear, affine, and ordered types. These extensions require eliminating or restricting the !_\tau expression. In such circumstances the type operator is not a true cartesian product, and is generally written to make this clear.
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She won a scholarship to study at Somerville College, Oxford, where she concentrated on topology in her mathematical studies with Henry Whitehead, earning a second-class degree in 1953. She earned her doctorate at the University of Warsaw, with a dissertation titled "The Homology of Cartesian Product Spaces" (1966).
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In mathematics, a ternary relation or triadic relation is a finitary relation in which the number of places in the relation is three. Ternary relations may also be referred to as 3-adic, 3-ary, 3-dimensional, or 3-place. Just as a binary relation is formally defined as a set of pairs, i.e. a subset of the Cartesian product of some sets A and B, so a ternary relation is a set of triples, forming a subset of the Cartesian product of three sets A, B and C. An example of a ternary relation in elementary geometry can be given on triples of points, where a triple is in the relation if the three points are collinear.
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The tool finds abstract test cases by calculating a finite model for each leaf in a testing tree . Finite models are calculated by restricting the type of each VIS variable to a finite set and then by calculating the Cartesian product between these sets. Each leaf predicate is evaluated on each element of this Cartesian product until one satisfies the predicate (meaning that an abstract test case was found) or until it is exhausted (meaning that either the test class is unsatisfiable or the finite model is inadequate). In the last case, the user has the chance to assist the tool in finding the right finite model or to prune the test class because it is unsatisfiable.
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Pisanski’s research interests span several areas of discrete and computational mathematics, including combinatorial configurations, abstract polytopes, maps on surfaces, chemical graph theory, and the history of mathematics and science. In 1980 he calculated the genus of the Cartesian product of any pair of connected, bipartite, d-valent graphs using a method that was later called the White–Pisanski method.J.L. Gross and T.W. Tucker, Topological graph theory, Wiley Interscience, 1987 In 1982 Vladimir Batagelj and Pisanski proved that the Cartesian product of a tree and a cycle is Hamiltonian if and only if no degree of the tree exceeds the length of the cycle. They also proposed a conjecture concerning cyclic Hamiltonicity of graphs.
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The complex torus associated to a lattice spanned by two periods, ω1 and ω2. Corresponding edges are identified. In mathematics, a complex torus is a particular kind of complex manifold M whose underlying smooth manifold is a torus in the usual sense (i.e. the cartesian product of some number N circles).
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In particular, we can take the homomorphic image of an algebra, h(A). A subalgebra of A is a subset of A that is closed under all the operations of A. A product of some set of algebraic structures is the cartesian product of the sets with the operations defined coordinatewise.
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The name is due to E. C. Zeeman, who observed that any contractible 2-complex (such as the dunce hat) after taking the Cartesian product with the closed unit interval seemed to be collapsible. This observation became known as the Zeeman conjecture and was shown by Zeeman to imply the Poincaré conjecture.
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In computer science, X + Y sorting is the problem of sorting pairs of numbers by their sum. Given two finite sets and , both of the same length, the problem is to order all pairs in the Cartesian product in numerical order by the value of . The problem is attributed to Elwyn Berlekamp.
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If is the semidirect product of the normal subgroup and the subgroup , and both and are finite, then the order of equals the product of the orders of and . This follows from the fact that is of the same order as the outer semidirect product of and , whose underlying set is the Cartesian product .
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In a Cartesian closed category, the exponential operation can be used to raise an arbitrary object to the power of another object. This generalizes the Cartesian product in the category of sets. If 0 is an initial object in a Cartesian closed category, then the exponential object 00 is isomorphic to any terminal object 1.
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Moreover, these terms are also commonly used for a finite section of the infinite graph, as in "an 8×8 square grid". The term lattice graph has also been given in the literature to various other kinds of graphs with some regular structure, such as the Cartesian product of a number of complete graphs.
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Consider graphs G and H. If both G and H admit perfect state transfer at time t, then their Cartesian product G \, \square \, H admits perfect state transfer at time t. If either G or H admits perfect state transfer at time t, then their disjoint union G \sqcup H admits perfect state transfer at time t.
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There are thus no zero objects in Ord. The categorical product in Ord is given by the product order on the cartesian product. We have a forgetful functor Ord → Set that assigns to each preordered set the underlying set, and to each order-preserving function the underlying function. This functor is faithful, and therefore Ord is a concrete category.
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In ring theory, a branch of mathematics, a semisimple algebra is an associative artinian algebra over a field which has trivial Jacobson radical (only the zero element of the algebra is in the Jacobson radical). If the algebra is finite-dimensional this is equivalent to saying that it can be expressed as a Cartesian product of simple subalgebras.
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In mathematics, a binary function (also called bivariate function, or function of two variables) is a function that takes two inputs. Precisely stated, a function f is binary if there exists sets X, Y, Z such that :\,f \colon X \times Y \rightarrow Z where X \times Y is the Cartesian product of X and Y.
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There are many ways to construct orders out of given orders. The dual order is one example. Another important construction is the cartesian product of two partially ordered sets, taken together with the product order on pairs of elements. The ordering is defined by (a, x) ≤ (b, y) if (and only if) a ≤ b and x ≤ y.
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Then, partitioning the cycle into two paths between the endpoints of this first chord, a second chord is needed to prevent the two paths of this second partition from being induced. Because Meyniel graphs are perfect graphs, parity graphs are also perfect. They are exactly the graphs whose Cartesian product with a single edge remains perfect..
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To compute the q-relaxed intersection of m boxes of R^{n}, we project all m boxes with respect to the n axes. For each of the n groups of m intervals, we compute the q-relaxed intersection. We return Cartesian product of the n resulting intervals. Figure 3 provides an illustration of the 4-relaxed intersection of 6 boxes.
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In five and higher dimensions, there are 3 regular polytopes, the hypercube, simplex and cross-polytope. They are generalisations of the three-dimensional cube, tetrahedron and octahedron, respectively. There are no regular star polytopes in these dimensions. Most uniform higher-dimensional polytopes are obtained by modifying the regular polytopes, or by taking the Cartesian product of polytopes of lower dimensions.
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Together, all classifications form the classification tree. For semantic purpose, classifications can be grouped into compositions. The maximum number of test cases is the Cartesian product of all classes of all classifications in the tree, quickly resulting in large numbers for realistic test problems. The minimum number of test cases is the number of classes in the classification with the most containing classes.
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It will be shown that , and hence a one-to-one correspondence between and the group exists. For , let be the subset of consisting of all subsets of cardinality exactly . Then is the disjoint union of the . The number of subsets of of cardinality is at most because every subset with elements is an element of the -fold cartesian product of .
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For, the product is given by the Cartesian product, whereas the coproduct is given by the disjoint union. This category does not have a zero object. Block matrix algebra relies upon biproducts in categories of matrices.H.D. Macedo, J.N. Oliveira, Typing linear algebra: A biproduct-oriented approach, Science of Computer Programming, Volume 78, Issue 11, 1 November 2013, Pages 2160-2191, , .
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Thus, the pullback can be thought of as the ordinary (Cartesian) product, but with additional structure. Instead of "forgetting" , , and , one can also "trivialize" them by specializing to be the terminal object (assuming it exists). and are then uniquely determined and thus carry no information, and the pullback of this cospan can be seen to be the product of and .
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In 4D, continuous or discrete rotational symmetry about a plane corresponds to corresponding 2D rotational symmetry in every perpendicular plane, about the point of intersection. An object can also have rotational symmetry about two perpendicular planes, e.g. if it is the Cartesian product of two rotationally symmetry 2D figures, as in the case of e.g. the duocylinder and various regular duoprisms.
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In mathematics, a surface bundle over the circle is a fiber bundle with base space a circle, and with fiber space a surface. Therefore the total space has dimension 2 + 1 = 3. In general, fiber bundles over the circle are a special case of mapping tori. Here is the construction: take the Cartesian product of a surface with the unit interval.
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In many contexts, an n-dimensional interval is defined as a subset of \R^n that is the Cartesian product of n intervals, I = I_1\times I_2 \times \cdots \times I_n, one on each coordinate axis. For n=2, this can be thought of as region bounded by a square or rectangle, whose sides are parallel to the coordinate axes, depending on whether the width of the intervals are the same or not; likewise, for n=3, this can be thought of as a region bounded by an axis-aligned cube or a rectangular cuboid. In higher dimensions, the Cartesian product of n intervals is bounded by an n-dimensional hypercube or hyperrectangle. A facet of such an interval I is the result of replacing any non-degenerate interval factor I_k by a degenerate interval consisting of a finite endpoint of I_k.
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To prove that Tychonoff's theorem in its general version implies the axiom of choice, we establish that every infinite cartesian product of non-empty sets is nonempty. The trickiest part of the proof is introducing the right topology. The right topology, as it turns out, is the cofinite topology with a small twist. It turns out that every set given this topology automatically becomes a compact space.
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Relational algebra received little attention outside of pure mathematics until the publication of E.F. Codd's relational model of data in 1970. Codd proposed such an algebra as a basis for database query languages. (See section Implementations.) Five primitive operators of Codd's algebra are the selection, the projection, the Cartesian product (also called the cross product or cross join), the set union, and the set difference.
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In the mathematical field of graph theory, the ladder graph Ln is a planar undirected graph with 2n vertices and 3n-2 edges. The ladder graph can be obtained as the Cartesian product of two path graphs, one of which has only one edge: Ln,1 = Pn × P2.Hosoya, H. and Harary, F. "On the Matching Properties of Three Fence Graphs." J. Math. Chem.
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In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one talks about the product in category theory, which formalizes these notions. Examples are the product of sets, groups (described below), rings, and other algebraic structures.
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It is decidable in polynomial-time whether an UFA's language is a subset of another UFA's language. Let A and B be two UFAs. Let L(A) and L(B) be the languages accepted by those automata. Then L(A)⊆L(B) if and only if L(A∩B)=L(A), where A∩B denotes the Cartesian product automaton, which can be proven to be also unambiguous.
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For example, if f is the identity map (i.e., the map which fixes every point of the torus) then the resulting torus bundle M(f) is the three-torus: the Cartesian product of three circles. Seeing the possible kinds of torus bundles in more detail requires an understanding of William Thurston's geometrization program. Briefly, if f is finite order, then the manifold M(f) has Euclidean geometry.
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One of many ways to express the axiom of choice is to say that it is equivalent to the statement that the Cartesian product of a collection of non-empty sets is non-empty. The proof that this is equivalent to the statement of the axiom in terms of choice functions is immediate: one needs only to pick an element from each set to find a representative in the product. Conversely, a representative of the product is a set which contains exactly one element from each component. The axiom of choice occurs again in the study of (topological) product spaces; for example, Tychonoff's theorem on compact sets is a more complex and subtle example of a statement that is equivalent to the axiom of choice, and shows why the product topology may be considered the more useful topology to put on a Cartesian product.
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The generalization of the Klein bottle to higher genus is given in the article on the fundamental polygon. In another order of ideas, constructing 3-manifolds, it is known that a solid Klein bottle is homeomorphic to the Cartesian product of a Möbius strip and a closed interval. The solid Klein bottle is the non-orientable version of the solid torus, equivalent to D^2\times S^1.
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In geometry of 4 dimensions or higher, a proprism is a polytope resulting from the Cartesian product of two or more polytopes, each of two dimensions or higher. The term was coined by John Horton Conway for product prism. The dimension of the space of a proprism equals the sum of the dimensions of all its product elements. Proprisms are often seen as k-face elements of uniform polytopes.
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The Cartesian products of projective Hilbert spaces is not a projective space. Segre mapping is an embedding of the Cartesian product of two projective spaces into their tensor product. In quantum theory, it describes how to make states of the composite system from states of its constituents. It is only an embedding not a surjection; most of the tensor product space does not lie in its range and represents entangled states.
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Standard 52-card deck An illustrative example is the standard 52-card deck. The standard playing card ranks {A, K, Q, J, 10, 9, 8, 7, 6, 5, 4, 3, 2} form a 13-element set. The card suits form a four-element set. The Cartesian product of these sets returns a 52-element set consisting of 52 ordered pairs, which correspond to all 52 possible playing cards.
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Thus each F•-structure is an F-structure with one element distinguished. Pointing is related to differentiation by the relation F• = X·F' , so F•(x) = x F' (x). The species of pointed sets, E•, is particularly important as a building block for many of the more complex constructions. The Cartesian product of two species is a species which can build two structures on the same set at the same time.
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In category theory, the product of two (or more) objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the Cartesian product of sets, the direct product of groups or rings, and the product of topological spaces. Essentially, the product of a family of objects is the "most general" object which admits a morphism to each of the given objects.
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Every connected graph in which the domination number is half the number of vertices arises in this way, with the exception of the four-vertex cycle graph. These graphs can be used to generate examples in which the bound of Vizing's conjecture, an unproven inequality between the domination number of the graphs in a different graph product, the cartesian product of graphs, is exactly met . They are also well-covered graphs.
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In mathematics, a compact semigroup is a semigroup in which the sets of solutions to equations can be described by finite sets of equations. The term "compact" here does not refer to any topology on the semigroup. Let S be a semigroup and X a finite set of letters. A system of equations is a subset E of the Cartesian product X∗ × X∗ of the free monoid (finite strings) over X with itself.
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When \otimes is the Cartesian product \times, the object Y \Rightarrow Z is called the exponential object, and is often written as Z^Y. Internal Homs, when chained together, form a language, called the internal language of the category. The most famous of these are simply typed lambda calculus, which is the internal language of Cartesian closed categories, and the linear type system, which is the internal language of closed symmetric monoidal categories.
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A record can be viewed as the computer analog of a mathematical tuple, although a tuple may or may not be considered a record, and vice versa, depending on conventions and the specific programming language. In the same vein, a record type can be viewed as the computer language analog of the Cartesian product of two or more mathematical sets, or the implementation of an abstract product type in a specific language.
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Zaslow 2008 A torus is the cartesian product of two circles. One approach to understanding mirror symmetry is the SYZ conjecture, which was suggested by Andrew Strominger, Shing-Tung Yau, and Eric Zaslow in 1996.Strominger, Yau, and Zaslow 1996 According to the SYZ conjecture, mirror symmetry can be understood by dividing a complicated Calabi-Yau manifold into simpler pieces and considering the effects of T-duality on these pieces.Yau and Nadis 2010, p.
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Prism graphs are examples of generalized Petersen graphs, with parameters GP(n,1). They may also be constructed as the Cartesian product of a cycle graph with a single edge. As with many vertex-transitive graphs, the prism graphs may also be constructed as Cayley graphs. The order-n dihedral group is the group of symmetries of a regular n-gon in the plane; it acts on the n-gon by rotations and reflections.
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Starting from a cyclically ordered set , one may form a linear order by unrolling it along an infinite line. This captures the intuitive notion of keeping track of how many times one goes around the circle. Formally, one defines a linear order on the Cartesian product , where is the set of integers, by fixing an element and requiring that for all :; :If , then . For example, the months , , , and occur in that order.
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A SystemVerilog coverage group creates a database of "bins" that store a histogram of values of an associated variable. Cross-coverage can also be defined, which creates a histogram representing the Cartesian product of multiple variables. A sampling event controls when a sample is taken. The sampling event can be a Verilog event, the entry or exit of a block of code, or a call to the `sample` method of the coverage group.
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The result is a partial order. If and are each totally ordered, then the result is a total order as well. The lexicographical order of two totally ordered sets is thus a linear extension of their product order. One can define similarly the lexicographic order on the Cartesian product of an infinite family of ordered sets, if the family is indexed by the nonnegative integers, or more generally by a well-ordered set.
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A right triangular prism is semiregular or, more generally, a uniform polyhedron if the base faces are equilateral triangles, and the other three faces are squares. It can be seen as a truncated trigonal hosohedron, represented by Schläfli symbol t{2,3}. Alternately it can be seen as the Cartesian product of a triangle and a line segment, and represented by the product {3}x{}. The dual of a triangular prism is a triangular bipyramid.
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For example, let as a subposet of the real numbers. The subset is a bounded interval, but it has no infimum or supremum in P, so it cannot be written in interval notation using elements of P. A poset is called locally finite if every bounded interval is finite. For example, the integers are locally finite under their natural ordering. The lexicographical order on the cartesian product ℕ×ℕ is not locally finite, since .
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On the Cartesian product of two sets with binary relations R and S, define (a, b)T(c, d) as aRc and bSd. If R and S are both reflexive, irreflexive, transitive, symmetric, or antisymmetric, then T will be also.Equivalence and Order Combining properties it follows that this also applies for being a preorder and being an equivalence relation. However if R and S are total relations, T is in not general total.
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A pattern is a discernible regularity in the world or in a manmade design. As such, the elements of a pattern repeat in a predictable manner. A geometric pattern is a kind of pattern formed of geometric shapes and typically repeating like a wallpaper. A relation on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2 = .
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Shed Skin combines Ole Agesen's Cartesian Product Algorithm (CPA) with the data-polymorphic part of John Plevyak's Iterative Flow Analysis (IFA).Master Thesis Mark Dufour, "Shed Skin. An Optimizing Python-to-C++ Compiler", April 19, 2006 Version 0.6 introduced scalability improvements which repeatedly analyze larger versions of a program (in addition to the mentioned techniques), until it is fully analyzed. This allows Shed Skin to do type inference on larger programs than previously.
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The 'plane' formed by a finite field is the cartesian product of all ordered pairs of field elements, with opposite edges identified forming the surface topologically equivalent to a discretized torus. Individual elements correspond to standard 'points' and 'lines' to sets of no more than p points related by incidence (an initial point) plus direction or slope given in lowest terms (say all points '2 over and 1 up') that 'wrap' the plane before repeating.
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The modular product of graphs. In graph theory, the modular product of graphs G and H is a graph formed by combining G and H that has applications to subgraph isomorphism. It is one of several different kinds of graph products that have been studied, generally using the same vertex set (the Cartesian product of the sets of vertices of the two graphs G and H) but with different rules for determining which edges to include.
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The key idea in tacit programming is to assist in operating at the appropriate level of abstraction. That is, to translate the natural transformation given by currying : \hom(A \times B, C) \cong \hom(B, C^A) into computer functions, where the left represents the uncurried form of a function and the right the curried. CA denotes the functionals from A to C (see also exponential object), while A × B denotes the Cartesian product of A and B.
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This problem can be solved using a straightforward comparison sort on the Cartesian product of the two given sets. When the sets have size n, their product has size n^2, and the time for a comparison sorting algorithm is O(n^2\log n). This is the best upper bound known on the time for this problem. Whether X + Y sorting can be done in a more slowly-growing time bound is an open problem.
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A stereographic projection of a Clifford torus performing a simple rotation Topologically a rectangle is the fundamental polygon of a torus, with opposite edges sewn together. In geometric topology, the Clifford torus is the simplest and most symmetric flat embedding of the cartesian product of two circles S and S (in the same sense that the surface of a cylinder is "flat"). It is named after William Kingdon Clifford. It resides in R4, as opposed to in R3.
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In this case, we start with bases uα and wβ of L/K and M/L respectively, where α is taken from an indexing set A, and β from an indexing set B. Using an entirely similar argument as the one above, we find that the products uαwβ form a basis for M/K. These are indexed by the cartesian product A × B, which by definition has cardinality equal to the product of the cardinalities of A and B.
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This class will then in turn be derived into multiple platform-specific implementations, each with platform-specific operations. Hence, only one set of `Window` classes are needed for each type of `Window`, and only one set of `WindowImp` classes are needed for each platform (rather than the Cartesian product of all available types and platforms). In addition, adding a new window type does not require any modification of platform implementation, or vice versa. This is a Bridge pattern.
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In topology, the cartesian product of topological spaces can be given several different topologies. One of the more obvious choices is the box topology, where a base is given by the Cartesian products of open sets in the component spaces.Willard, 8.2 pp. 52-53, Another possibility is the product topology, where a base is given by the Cartesian products of open sets in the component spaces, only finitely many of which can be not equal to the entire component space.
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A cube is a collection of cells, each of which is identified by a tuple of elements, one from each dimension of the cube. Each cell in a cube contains a value. A cube is effectively a function that assigns a value to each n-tuple of the cartesian product of the dimensions. The value of a cell may be assigned externally (input), or the result of a calculation that uses other cells in the same cube or other cubes.
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In 1959, he published his classical monograph, Theory of Value: An Axiomatic Analysis of Economic Equilibrium (Cowles Foundation Monographs Series), which is one of the most important works in mathematical economics. He also studied several problems in the theory of cardinal utility, in particular the additive decomposition of a utility function defined on a Cartesian product of sets. In this monograph, Debreu set up an axiomatic foundation for competitive markets. He also established the existence of an equilibrium using a novel approach.
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If faces are all regular, the hexagonal prism is a semiregular polyhedron, more generally, a uniform polyhedron, and the fourth in an infinite set of prisms formed by square sides and two regular polygon caps. It can be seen as a truncated hexagonal hosohedron, represented by Schläfli symbol t{2,6}. Alternately it can be seen as the Cartesian product of a regular hexagon and a line segment, and represented by the product {6}×{}. The dual of a hexagonal prism is a hexagonal bipyramid.
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Classical kinematics does not primarily demand experimental description of its phenomena. It allows completely precise description of an instantaneous state by a value in phase space, the Cartesian product of configuration and momentum spaces. This description simply assumes or imagines a state as a physically existing entity without concern about its experimental measurability. Such a description of an initial condition, together with Newton's laws of motion, allows a precise deterministic and causal prediction of a final condition, with a definite trajectory of passage.
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If faces are all regular, the pentagonal prism is a semiregular polyhedron, more generally, a uniform polyhedron, and the third in an infinite set of prisms formed by square sides and two regular polygon caps. It can be seen as a truncated pentagonal hosohedron, represented by Schläfli symbol t{2,5}. Alternately it can be seen as the Cartesian product of a regular pentagon and a line segment, and represented by the product {5}x{}. The dual of a pentagonal prism is a pentagonal bipyramid.
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The holonomy was introduced by in order to study and classify symmetric spaces. It was not until much later that holonomy groups would be used to study Riemannian geometry in a more general setting. In 1952 Georges de Rham proved the de Rham decomposition theorem, a principle for splitting a Riemannian manifold into a Cartesian product of Riemannian manifolds by splitting the tangent bundle into irreducible spaces under the action of the local holonomy groups. Later, in 1953, Marcel Berger classified the possible irreducible holonomies.
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The ridge of the duocylinder is the 2-manifold that is the boundary between the two bounding (solid) torus cells. It is in the shape of a Clifford torus, which is the Cartesian product of two circles. Intuitively, it may be constructed as follows: Roll a 2-dimensional rectangle into a cylinder, so that its top and bottom edges meet. Then roll the cylinder in the plane perpendicular to the 3-dimensional hyperplane that the cylinder lies in, so that its two circular ends meet.
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Namioka's book Linear Topological Spaces with Kelley has become a "standard text". Although his doctoral work and this book both concerned general topology, his interests later shifted to functional analysis.. With Asplund in 1967, Namioka gave one of the first complete proofs of the Ryll-Nardzewski fixed-point theorem.. Following his 1974 paper "separate continuity and joint continuity", a Namioka space has come to mean a topological space X with the property that whenever Y is a compact space and function f from the Cartesian product of X and Y to Z is separately continuous in X and Y, there must exist a dense Gδ set within X whose Cartesian product with Y is a subset of the set of points of continuity of f.. The result of the 1974 paper, a proof of this property for a specific class of topological spaces, has come to be known as Namioka's theorem. In 1975, Namioka and Phelps established one side of the theorem that a space is an Asplund space if and only if its dual space has the Radon–Nikodým property. The other side was completed in 1978 by Stegall..
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The three- dimensional cube and its dual, the octahedron, the two three-dimensional Hanner polytopes Schlegel diagram of the octahedral prism A cube is a Hanner polytope, and can be constructed as a Cartesian product of three line segments. Its dual, the octahedron, is also a Hanner polytope, the direct sum of three line segments. In three dimensions all Hanner polytopes are combinatorially equivalent to one of these two types of polytopes.. In higher dimensions the hypercubes and cross polytopes, analogues of the cube and octahedron, are again Hanner polytopes. However, more examples are possible.
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A Cartesian product of a family of topological vector spaces, when endowed with the product topology, is a topological vector space. For instance, the set of all functions : this set can be identified with the product space and carries a natural product topology. With this topology, becomes a topological vector space, endowed with a topology called the topology of pointwise convergence. The reason for this name is the following: if is a sequence of elements in , then has limit if and only if has limit for every real number x.
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In graph theory, Vizing's conjecture concerns a relation between the domination number and the cartesian product of graphs. This conjecture was first stated by , and states that, if γ(G) denotes the minimum number of vertices in a dominating set for G, then : \gamma(G\,\Box\,H) \ge \gamma(G)\gamma(H). \, conjectured a similar bound for the domination number of the tensor product of graphs; however, a counterexample was found by . Since Vizing proposed his conjecture, many mathematicians have worked on it, with partial results described below.
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The Cartesian product structure of on Cartesian plane of ordered pairs . Blue lines denote coordinate axes, horizontal green lines are integer , vertical cyan lines are integer , brown-orange lines show half-integer or , magenta and its tint show multiples of one tenth (best seen under magnification) In mathematics, a real coordinate space of dimension , written ( ) or , is a coordinate space over the real numbers. This means that it is the set of the -tuples of real numbers (sequences of real numbers). With component-wise addition and scalar multiplication, it is a real vector space.
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Any kind of interval, open, closed, semi-open, semi-closed, open-bounded, compact, even rays, can be the fiber. Two simple examples of I-bundles are the annulus and the Möbius band, the only two possible I-bundles over the circle S^1. The annulus is a trivial or untwisted bundle because it corresponds to the Cartesian product S^1\times I, and the Möbius band is a non-trivial or twisted bundle. Both bundles are 2-manifolds, but the annulus is an orientable manifold while the Möbius band is a non-orientable manifold.
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Compactly generated spaces were originally called k-spaces, after the German word kompakt. They were studied by Hurewicz, and can be found in General Topology by Kelley, Topology by Dugundji, Rational Homotopy Theory by Félix, Halperin, and Thomas. The motivation for their deeper study came in the 1960s from well known deficiencies of the usual topological category. This fails to be a cartesian closed category, the usual cartesian product of identification maps is not always an identification map, and the usual product of CW-complexes need not be a CW-complex.
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The product of a finite set of metric spaces in Met is a metric space that has the cartesian product of the spaces as its points; the distance in the product space is given by the supremum of the distances in the base spaces. That is, it is the product metric with the sup norm. However, the product of an infinite set of metric spaces may not exist, because the distances in the base spaces may not have a supremum. That is, Met is not a complete category, but it is finitely complete.
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The principal homogeneous space concept is a special case of that of principal bundle: it means a principal bundle with base a single point. In other words the local theory of principal bundles is that of a family of principal homogeneous spaces depending on some parameters in the base. The 'origin' can be supplied by a section of the bundle--such sections are usually assumed to exist locally on the base--the bundle being locally trivial, so that the local structure is that of a cartesian product. But sections will often not exist globally.
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For each such subset, the mean is computed; finally, the median of these n(n + 1)/2 averages is defined to be the Hodges–Lehmann estimator of location. The Hodges–Lehmann statistic also estimates the difference between two populations. For two sets of data with m and n observations, the set of two-element sets made of them is their Cartesian product, which contains m × n pairs of points (one from each set); each such pair defines one difference of values. The Hodges–Lehmann statistic is the median of the m × n differences.
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Ordinary monoids are precisely the monoid objects in the cartesian monoidal category Set. Further, any strict monoidal category can be seen as a monoid object in the category of categories Cat (equipped with the monoidal structure induced by the cartesian product). Monoidal functors are the functors between monoidal categories that preserve the tensor product and monoidal natural transformations are the natural transformations, between those functors, which are "compatible" with the tensor product. Every monoidal category can be seen as the category B(∗, ∗) of a bicategory B with only one object, denoted ∗.
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The Hartley function only depends on the number of elements in a set, and hence can be viewed as a function on natural numbers. Rényi showed that the Hartley function in base 2 is the only function mapping natural numbers to real numbers that satisfies # H(mn) = H(m)+H(n) (additivity) # H(m) \leq H(m+1) (monotonicity) # H(2)=1 (normalization) Condition 1 says that the uncertainty of the Cartesian product of two finite sets A and B is the sum of uncertainties of A and B. Condition 2 says that a larger set has larger uncertainty.
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Currying is most easily understood by starting with an informal definition, which can then be molded to fit many different domains. First, there is some notation to be established. The notation X \to Y denotes all functions from X to Y. If f is such a function, we write f \colon X \to Y . Let X \times Y denote the ordered pairs of the elements of X and Y respectively, that is, the Cartesian product of X and Y. Here, X and Y may be sets, or they may be types, or they may be other kinds of objects, as explored below.
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A simple example of a smooth fiber bundle is a Cartesian product of two manifolds. Consider the bundle B1 := (M × N, pr1) with bundle projection pr1 : M × N → M : (x, y) → x. Applying the definition in the paragraph above to find the vertical bundle, we consider first a point (m,n) in M × N. Then the image of this point under pr1 is m. The preimage of m under this same pr1 is {m} × N, so that T(m,n) ({m} × N) = {m} × TN. The vertical bundle is then VB1 = M × TN, which is a subbundle of T(M ×N).
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In algebraic geometry, a correspondence between algebraic varieties V and W is a subset R of V×W, that is closed in the Zariski topology. In set theory, a subset of a Cartesian product of two sets is called a binary relation or correspondence; thus, a correspondence here is a relation that is defined by algebraic equations. There are some important examples, even when V and W are algebraic curves: for example the Hecke operators of modular form theory may be considered as correspondences of modular curves. However, the definition of a correspondence in algebraic geometry is not completely standard.
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Hence, a magma is medial if and only if its binary operation is a magma homomorphism from to . This can easily be expressed in terms of a commutative diagram, and thus leads to the notion of a medial magma object in a category with a Cartesian product. (See the discussion in auto magma object.) If and are endomorphisms of a medial magma, then the mapping defined by pointwise multiplication :(f\cdot g)(x) = f(x)\cdot g(x) is itself an endomorphism. It follows that the set End() of all endomorphisms of a medial magma is itself a medial magma.
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A grid graph is a unit distance graph corresponding to the square lattice, so that it is isomorphic to the graph having a vertex corresponding to every pair of integers (a, b), and an edge connecting (a, b) to (a+1, b) and (a, b+1). The finite grid graph Gm,n is an m×n rectangular graph isomorphic to the one obtained by restricting the ordered pairs to the range 0 ≤ a < m, 0 ≤ b < n. Grid graphs can be obtained as the Cartesian product of two paths: Gm,n = Pm × Pn. Every grid graph is a median graph.
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Solid torus In mathematics, a solid torus is the topological space formed by sweeping a disk around a circle.. It is homeomorphic to the Cartesian product S^1 \times D^2 of the disk and the circle,. endowed with the product topology. A standard way to visualize a solid torus is as a toroid, embedded in 3-space. However, it should be distinguished from a torus, which has the same visual appearance: the torus is the two-dimensional space on the boundary of a toroid, while the solid torus includes also the compact interior space enclosed by the torus.
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Eval and apply are the two interdependent components of the eval-apply cycle, which is the essence of evaluating Lisp, described in SICP.The Metacircular Evaluator (SICP Section 4.1) In category theory, the eval morphism is used to define the closed monoidal category. Thus, for example, the category of sets, with functions taken as morphisms, and the cartesian product taken as the product, forms a Cartesian closed category. Here, eval (or, properly speaking, apply) together with its right adjoint, currying, form the simply typed lambda calculus, which can be interpreted to be the morphisms of Cartesian closed categories.
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By repeatedly removing leaves in this way, the Wiener index may be calculated in linear time.. For graphs that are constructed as products of simpler graphs, the Wiener index of the product graph can often be computed by a simple formula that combines the indices of its factors.. Benzenoids (graphs formed by gluing regular hexagons edge-to- edge) can be embedded isometrically into the Cartesian product of three trees, allowing their Wiener indices to be computed in linear time by using the product formula together with the linear time tree algorithm.. For earlier algorithms for benzenoids based on their partial cube structure, see .
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For example, like sequences, nets can be "plugged into" other functions, where "plugging in" is just function composition. Theorems related to functions and function composition may then be applied to nets. One example is the universal property of inverse limits, which is defined in terms of composition of maps rather than sets and it is more readily applied to functions like nets than to sets like filters (a prominent examples of an inverse limit is the Cartesian product). In contrast to nets, filters (and more generally prefilters) are families of __sets__ and so they have the advantages of sets.
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Given a finite group G, the generators of its Burnside ring Ω(G) are the formal differences of isomorphism classes of finite G-sets. For the ring structure, addition is given by disjoint union of G-sets and multiplication by their Cartesian product. The Burnside ring is a free Z-module, whose generators are the (isomorphism classes of) orbit types of G. If G acts on a finite set X, then one can write X = \bigcup_i X_i (disjoint union), where each Xi is a single G-orbit. Choosing any element xi in Xi creates an isomorphism G/Gi → Xi, where Gi is the stabilizer (isotropy) subgroup of G at xi.
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In mathematics, specifically in group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted . This operation is the group-theoretic analogue of the Cartesian product of sets and is one of several important notions of direct product in mathematics. In the context of abelian groups, the direct product is sometimes referred to as the direct sum, and is denoted G \oplus H. Direct sums play an important role in the classification of abelian groups: according to the fundamental theorem of finite abelian groups, every finite abelian group can be expressed as the direct sum of cyclic groups.
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In mathematics, a heterogeneous relation is a binary relation, a subset of a Cartesian product A × B, where A and B are distinct sets. The prefix hetero is from the Greek ἕτερος (heteros, "other, another, different"). A heterogeneous relation has been called a rectangular relation, suggesting that it does not have the square-symmetry of a homogeneous relation on a set where A = B. Commenting on the development of binary relations beyond homogeneous relations, researchers wrote, "...a variant of the theory has evolved that treats relations from the very beginning as heterogeneous or rectangular, i.e. as relations where the normal case is that they are relations between different sets."G.
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Once we have this fact, Tychonoff's theorem can be applied; we then use the finite intersection property (FIP) definition of compactness. The proof itself (due to J. L. Kelley) follows: Let {Ai} be an indexed family of nonempty sets, for i ranging in I (where I is an arbitrary indexing set). We wish to show that the cartesian product of these sets is nonempty. Now, for each i, take Xi to be Ai with the index i itself tacked on (renaming the indices using the disjoint union if necessary, we may assume that i is not a member of Ai, so simply take Xi = Ai ∪ {i}).
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The topological structure of (called standard topology, Euclidean topology, or usual topology) can be obtained not only from Cartesian product. It is also identical to the natural topology induced by Euclidean metric discussed above: a set is open in the Euclidean topology if and only if it contains an open ball around each of its points. Also, is a linear topological space (see continuity of linear maps above), and there is only one possible (non-trivial) topology compatible with its linear structure. As there are many open linear maps from to itself which are not isometries, there can be many Euclidean structures on which correspond to the same topology.
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More generally, differential invariants can be considered for mappings from any smooth manifold X into another smooth manifold Y for a Lie group acting on the Cartesian product X×Y. The graph of a mapping X -> Y is a submanifold of X×Y that is everywhere transverse to the fibers over X. The group G acts, locally, on the space of such graphs, and induces an action on the k-th prolongation Y(k) consisting of graphs passing through each point modulo the relation of k-th order contact. A differential invariant is a function on Y(k) that is invariant under the prolongation of the group action.
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The frame bundle also comes equipped with a solder form which is horizontal in the sense that it vanishes on vertical vectors such as the point values of the vector fields : indeed is defined first by projecting a tangent vector (to at a frame ) to , then by taking the components of this tangent vector on with respect to the frame . Note that is also -equivariant (where acts on by matrix multiplication). The pair defines a bundle isomorphism of with the trivial bundle , where is the Cartesian product of and (viewed as the Lie algebra of the affine group, which is actually a semidirect product – see below).
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In mathematics, specifically in topology of manifolds, a compact codimension- one submanifold F of a manifold M is said to be 2-sided in M when there is an embedding ::h\colon F\times [-1,1]\to M with h(x,0)=x for each x\in F and ::h(F\times [-1,1])\cap \partial M=h(\partial F\times [-1,1]). In other words, if its normal bundle is trivial. This means, for example that a curve in a surface is 2-sided if it has a tubular neighborhood which is a cartesian product of the curve times an interval. A submanifold which is not 2-sided is called 1-sided.
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Any two skew lines of these 27 belong to a unique Schläfli double six configuration, a set of 12 lines whose intersection graph is a crown graph in which the two lines have disjoint neighborhoods. Correspondingly, in the Schläfli graph, each edge uv belongs uniquely to a subgraph in the form of a Cartesian product of complete graphs K6 \square K2 in such a way that u and v belong to different K6 subgraphs of the product. The Schläfli graph has a total of 36 subgraphs of this form, one of which consists of the zero-one vectors in the eight-dimensional representation described above.
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In group theory, Hajós's theorem states that if a finite abelian group is expressed as the Cartesian product of simplexes, that is, sets of the form {e,a,a2,...,as-1} where e is the identity element, then at least one of the factors is a subgroup. The theorem was proved by the Hungarian mathematician György Hajós in 1941 using group rings. Rédei later proved the statement when the factors are only required to contain the identity element and be of prime cardinality. In this lattice tiling of the plane by congruent squares, the green and violet squares meet edge-to-edge as do the blue and orange squares.
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Given two measured dynamical systems (X, \mu, T) and (Y, u, S), one can construct a dynamical system (X \times Y, \mu \otimes u, T \times S) on the Cartesian product by defining (T \times S) (x,y) = (T(x), S(y)). We then have the following characterizations of weak mixing: :Proposition. A dynamical system (X, \mu, T) is weakly mixing if and only if, for any ergodic dynamical system (Y, u, S), the system (X \times Y, \mu \otimes u, T \times S) is also ergodic. :Proposition. A dynamical system (X, \mu, T) is weakly mixing if and only if (X^2, \mu \otimes \mu, T \times T) is also ergodic.
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In type theory, the general idea of a type system in computer science is formalized into a specific algebra of types. For example, when writing f \colon X \to Y , the intent is that X and Y are types, while the arrow \to is a type constructor, specifically, the function type or arrow type. Similarly, the Cartesian product X \times Y of types is constructed by the product type constructor \times. The type-theoretical approach is expressed in programming languages such as ML and the languages derived from and inspired by it: CaML, Haskell and F#. The type-theoretical approach provides a natural complement to the language of category theory, as discussed below.
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Like Set, FinSet and FinOrd are topoi. As in Set, in FinSet the categorical product of two objects A and B is given by the cartesian product , the categorical sum is given by the disjoint union , and the exponential object BA is given by the set of all functions with domain A and codomain B. In FinOrd, the categorical product of two objects n and m is given by the ordinal product , the categorical sum is given by the ordinal sum , and the exponential object is given by the ordinal exponentiation nm. The subobject classifier in FinSet and FinOrd is the same as in Set. FinOrd is an example of a PRO.
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His postulates follow, to some degree Euclid in form, but the axiomatic ideas about sets issuing from the 19th and early 20th centuries, in content. Broadly analogous to Euclid's postulates on the construction of a circle given a point and a line or the construction of a unique straight line given two points are the postulates to do with Union, Power Set and Cartesian product which posit global constructions producing new arithmoi from one or more given ones. Somewhat different however are his postulates on Replacement and Comprehension. These do not set out individual constructions which simply have to be grasped but rather make affirmations about all possible constructions and all conceivable properties.
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An Algebraic Theory T is a category whose objects are natural numbers 0, 1, 2,..., and which, for each n, has an n-tuple of morphisms: proji: n → 1, i = 1,..., n This allows interpreting n as a cartesian product of n copies of 1. Example. Let's define an algebraic theory T taking hom(n, m) to be m-tuples of polynomials of n free variables X1,..., Xn with integer coefficients and with substitution as composition. In this case proji is the same as Xi. This theory T is called the theory of commutative rings. In an algebraic theory, any morphism n → m can be described as m morphisms of signature n → 1.
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The above result can be restated in a different way. For a semisimple algebra A = A1 ×...× An expressed in terms of its simple factors, consider the units ei ∈ Ai. The elements Ei = (0,...,ei,...,0) are idempotent elements in A and they lie in the center of A. Furthermore, Ei A = Ai, EiEj = 0 for i ≠ j, and Σ Ei = 1, the multiplicative identity in A. Therefore, for every semisimple algebra A, there exists idempotents {Ei} in the center of A, such that #EiEj = 0 for i ≠ j (such a set of idempotents is called central orthogonal), #Σ Ei = 1, #A is isomorphic to the Cartesian product of simple algebras E1 A ×...× En A.
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In mathematics, the n-fold symmetric product of an algebraic curve C is the quotient space of the n-fold cartesian product :C × C × ... × C or Cn by the group action of the symmetric group on n letters permuting the factors. It exists as a smooth algebraic variety ΣnC; if C is a compact Riemann surface it is therefore a complex manifold. Its interest in relation to the classical geometry of curves is that its points correspond to effective divisors on C of degree n, that is, formal sums of points with non-negative integer coefficients. For C the projective line (say the Riemann sphere) ΣnC can be identified with projective space of dimension n.
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A categorical product can be characterized by a universal construction. For concreteness, one may consider the Cartesian product in Set, the direct product in Grp, or the product topology in Top, where products exist. Let X and Y be objects of a category D with finite products. The product of X and Y is an object X × Y together with two morphisms :\pi_1 : X \times Y \to X :\pi_2 : X \times Y \to Y such that for any other object Z of D and morphisms f: Z \to X and g: Z \to Y there exists a unique morphism h: Z \to X \times Y such that f = \pi_1 \circ h and g = \pi_2 \circ h.
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The tensor product of two modules and over a commutative ring is defined in exactly the same way as the tensor product of vector spaces over a field: :A \otimes_R B := F (A \times B) / G where now is the free -module generated by the cartesian product and is the -module generated by the same relations as above. More generally, the tensor product can be defined even if the ring is non-commutative. In this case has to be a right--module and is a left--module, and instead of the last two relations above, the relation :(ar,b)-(a,rb) is imposed. If is non-commutative, this is no longer an -module, but just an abelian group.
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Operations preserving word-representability are removing a vertex, replacing a vertex with a module, Cartesian product, rooted product, subdivision of a graph, connecting two graphs by an edge and gluing two graphs in a vertex . The operations not necessarily preserving word-representability are taking the complement, taking the line graph, edge contraction , gluing two graphs in a clique of size 2 or more , tensor product, lexicographic product and strong product I. Choi, J. Kim, and M. Kim. On operations preserving semi-transitive orient ability of graphs, Journal of Combinatorial Optimization 37 (2019) 4, 1351−1366.. Edge-deletion, edge-addition and edge-lifting with respect to word-representability (equivalently, semi-transitive orientability) are studied in .
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Often, the use of the term operation implies that the domain of the function includes a power of the codomain (i.e. the Cartesian product of one or more copies of the codomain), although this is by no means universal, as in the case of dot product, where vectors are multiplied and result in a scalar. An n-ary operation is called an internal operation. An n-ary operation where is called an external operation by the scalar set or operator set S. In particular for a binary operation, is called a left-external operation by S, and is called a right-external operation by S. An example of an internal operation is vector addition, where two vectors are added and result in a vector.
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If is a map between topological spaces then the graph of is the set } or equivalently, :} We say that the graph of is closed if is a closed subset of (with the product topology). Any continuous function into a Hausdorff space has a closed graph. Any linear map, , between two topological vector spaces whose topologies are (Cauchy) complete with respect to translation invariant metrics, and if in addition (1a) is sequentially continuous in the sense of the product topology, then the map is continuous and its graph, , is necessarily closed. Conversely, if is such a linear map with, in place of (1a), the graph of is (1b) known to be closed in the Cartesian product space , then is continuous and therefore necessarily sequentially continuous.
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Now take S to be the set of sequences of k elements selected from our n-element set without repetition. On one hand, there is an easy bijection of S with the Cartesian product corresponding to the numerator n(n-1)\cdots(n-k+1), and on the other hand there is a bijection from the set C of pairs of a k-combination and a permutation σ of k to S, by taking the elements of C in increasing order, and then permuting this sequence by σ to obtain an element of S. The two ways of counting give the equation :n(n-1)\cdots(n-k+1)=\binom nk k!, and after division by k! this leads to the stated formula for \tbinom nk.
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To describe the above properties in the case where G is the direct sum of an infinite (perhaps uncountable) set of subgroups, more care is needed. If g is an element of the cartesian product ∏{Hi} of a set of groups, let gi be the ith element of g in the product. The external direct sum of a set of groups {Hi} (written as ∑E{Hi}) is the subset of ∏{Hi}, where, for each element g of ∑E{Hi}, gi is the identity e_{H_i} for all but a finite number of gi (equivalently, only a finite number of gi are not the identity). The group operation in the external direct sum is pointwise multiplication, as in the usual direct product.
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One infinite-dimensional generalization is as follows. Let and be Banach spaces, and a pair of open sets. Let :F:A\times B \to L(X,Y) be a continuously differentiable function of the Cartesian product (which inherits a differentiable structure from its inclusion into X × Y) into the space of continuous linear transformations of into Y. A differentiable mapping u : A → B is a solution of the differential equation :(1) \quad y' = F(x,y) if :\forall x \in A: \quad u'(x) = F(x, u(x)). The equation (1) is completely integrable if for each (x_0, y_0)\in A\times B, there is a neighborhood U of x0 such that (1) has a unique solution defined on U such that u(x0)=y0.
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To put the above informal discussion of braid groups on firm ground, one needs to use the homotopy concept of algebraic topology, defining braid groups as fundamental groups of a configuration space. Alternatively, one can define the braid group purely algebraically via the braid relations, keeping the pictures in mind only to guide the intuition. To explain how to reduce a braid group in the sense of Artin to a fundamental group, we consider a connected manifold X of dimension at least 2. The symmetric product of n copies of X means the quotient of X^n, the n-fold Cartesian product of X by the permutation action of the symmetric group on n strands operating on the indices of coordinates.
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When a group G acts on a set S, the action may be extended naturally to the Cartesian product Sn of S, consisting of n-tuples of elements of S: the action of an element g on the n-tuple (s1, ..., sn) is given by : g(s1, ..., sn) = (g(s1), ..., g(sn)). The group G is said to be oligomorphic if the action on Sn has only finitely many orbits for every positive integer n.Oligomorphic permutation groups - Isaac Newton Institute preprint, Peter J. Cameron (This is automatic if S is finite, so the term is typically of interest when S is infinite.) The interest in oligomorphic groups is partly based on their application to model theory, for example when considering automorphisms in countably categorical theories.
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To see why R4 is necessary, note that if S and S each exists in its own independent embedding space R and R, the resulting product space will be R4 rather than R3. The historically popular view that the cartesian product of two circles is an R3 torus in contrast requires the highly asymmetric application of a rotation operator to the second circle, since that circle will only have one independent axis z available to it after the first circle consumes x and y. Stated another way, a torus embedded in R3 is an asymmetric reduced-dimension projection of the maximally symmetric Clifford torus embedded in R4. The relationship is similar to that of projecting the edges of a cube onto a sheet of paper.
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The related truncated icosidodecahedral prism is constructed from two truncated icosidodecahedra connected by prisms, shown here in stereographic projection with some prisms hidden. The spherinder is related to the uniform prismatic polychora, which are cartesian product of a regular or semiregular polyhedron and a line segment. There are eighteen convex uniform prisms based on the Platonic and Archimedean solids (tetrahedral prism, truncated tetrahedral prism, cubic prism, cuboctahedral prism, octahedral prism, rhombicuboctahedral prism, truncated cubic prism, truncated octahedral prism, truncated cuboctahedral prism, snub cubic prism, dodecahedral prism, icosidodecahedral prism, icosahedral prism, truncated dodecahedral prism, rhombicosidodecahedral prism, truncated icosahedral prism, truncated icosidodecahedral prism, snub dodecahedral prism), plus an infinite family based on antiprisms, and another infinite family of uniform duoprisms, which are products of two regular polygons.
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By analogy, the subset of the Cartesian product X×X of any set X with itself, consisting of all pairs (x,x), is called the diagonal, and is the graph of the equality relation on X or equivalently the graph of the identity function from X to x. This plays an important part in geometry; for example, the fixed points of a mapping F from X to itself may be obtained by intersecting the graph of F with the diagonal. In geometric studies, the idea of intersecting the diagonal with itself is common, not directly, but by perturbing it within an equivalence class. This is related at a deep level with the Euler characteristic and the zeros of vector fields.
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Every Hanner polytope can be given vertex coordinates that are 0, 1, or −1.. More explicitly, if P and Q are Hanner polytopes with coordinates in this form, then the coordinates of the vertices of the Cartesian product of P and Q are formed by concatenating the coordinates of a vertex in P with the coordinates of a vertex in Q. The coordinates of the vertices of the direct sum of P and Q are formed either by concatenating the coordinates of a vertex in P with a vector of zeros, or by concatenating a vector of zeros with the coordinates of a vertex in Q. Because the polar dual of a Hanner polytope is another Hanner polytope, the Hanner polytopes have the property that both they and their duals have coordinates in {0,1,−1}.
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More precisely, a binary operation on a set S is a mapping of the elements of the Cartesian product to S: :\,f \colon S \times S \rightarrow S. Because the result of performing the operation on a pair of elements of S is again an element of S, the operation is called a closed (or internal) binary operation on S (or sometimes expressed as having the property of closure). If f is not a function, but is instead a partial function, it is called a partial binary operation. For instance, division of real numbers is a partial binary operation, because one can't divide by zero: a/0 is not defined for any real a. However, both in universal algebra and model theory the binary operations considered are defined on all of .
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One example of this is the Cartesian product, which is often seen as associative: :(E \times F) \times G = E \times (F \times G) = E \times F \times G. But this is strictly speaking not true: if x \in E, y \in F and z \in G, the identity ((x, y), z) = (x, (y, z)) would imply that (x, y) = x and z = (y, z), and so ((x, y), z) = (x, y, z) would mean nothing. However, these equalities can be legitimized and made rigorous in category theory—using the idea of a natural isomorphism. Another example of similar abuses occurs in statements such as "there are two non-Abelian groups of order 8", which more strictly stated means "there are two isomorphism classes of non-Abelian groups of order 8".
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For many experiments, there may be more than one plausible sample space available, depending on what result is of interest to the experimenter. For example, when drawing a card from a standard deck of fifty-two playing cards, one possibility for the sample space could be the various ranks (Ace through King), while another could be the suits (clubs, diamonds, hearts, or spades). A more complete description of outcomes, however, could specify both the denomination and the suit, and a sample space describing each individual card can be constructed as the Cartesian product of the two sample spaces noted above (this space would contain fifty-two equally likely outcomes). Still other sample spaces are possible, such as {right-side up, up-side down} if some cards have been flipped when shuffling.
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A set of rectangular ranges ('trellis') whose area has to be measured. In computational geometry, Klee's measure problem is the problem of determining how efficiently the measure of a union of (multidimensional) rectangular ranges can be computed. Here, a d-dimensional rectangular range is defined to be a Cartesian product of d intervals of real numbers, which is a subset of Rd. The problem is named after Victor Klee, who gave an algorithm for computing the length of a union of intervals (the case d = 1) which was later shown to be optimally efficient in the sense of computational complexity theory. The computational complexity of computing the area of a union of 2-dimensional rectangular ranges is now also known, but the case d ≥ 3 remains an open problem.
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To construct the Grothendieck group K of a commutative monoid M, one forms the Cartesian product M \times M. The two coordinates are meant to represent a positive part and a negative part, so (m_1, m_2) corresponds to m_1- m_2 in K. Addition on M\times M is defined coordinate-wise: :(m_1, m_2) + (n_1,n_2) = (m_1+n_1, m_2+n_2). Next one defines an equivalence relation on M \times M, such that (m_1, m_2) is equivalent to (n_1, n_2) if, for some element k of M, m1 \+ n2 \+ k = m2 \+ n1 \+ k (the element k is necessary because the cancellation law does not hold in all monoids). The equivalence class of the element (m1, m2) is denoted by [(m1, m2)]. One defines K to be the set of equivalence classes.
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The theory of combinatorial species and its extension to analytic combinatorics provide a language for describing many important combinatorial classes, constructing new classes from combinations of previously defined ones, and automatically deriving their counting sequences. For example, two combinatorial classes may be combined by disjoint union, or by a Cartesian product construction in which the objects are ordered pairs of one object from each of two classes, and the size function is the sum of the sizes of each object in the pair. These operations respectively form the addition and multiplication operations of a semiring on the family of (isomorphism equivalence classes of) combinatorial classes, in which the zero object is the empty combinatorial class, and the unit is the class whose only object is the empty set..
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In the context of first-order logic, the symbols in a signature are also known as the non-logical symbols, because together with the logical symbols they form the underlying alphabet over which two formal languages are inductively defined: The set of terms over the signature and the set of (well-formed) formulas over the signature. In a structure, an interpretation ties the function and relation symbols to mathematical objects that justify their names: The interpretation of an n-ary function symbol f in a structure A with domain A is a function fA: An → A, and the interpretation of an n-ary relation symbol is a relation RA ⊆ An. Here An = A × A × ... × A denotes the n-fold cartesian product of the domain A with itself, and so f is in fact an n-ary function, and R an n-ary relation.
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Simplex graphs were introduced by , credit the introduction of simplex graphs to a later paper, also by Bandelt and van de Vel, but this appears to be a mistake. who observed that a simplex graph has no cubes if and only if the underlying graph is triangle-free, and showed that the chromatic number of the underlying graph equals the minimum number n such that the simplex graph can be isometrically embedded into a Cartesian product of n trees. As a consequence of the existence of triangle-free graphs with high chromatic number, they showed that there exist two-dimensional topological median algebras that cannot be embedded into products of finitely many real trees. also use simplex graphs as part of their proof that testing whether a graph is triangle-free or whether it is a median graph may be performed equally quickly.
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In the mathematical fields of geometry and topology, a coarse structure on a set X is a collection of subsets of the cartesian product X × X with certain properties which allow the large-scale structure of metric spaces and topological spaces to be defined. The concern of traditional geometry and topology is with the small-scale structure of the space: properties such as the continuity of a function depend on whether the inverse images of small open sets, or neighborhoods, are themselves open. Large-scale properties of a space--such as boundedness, or the degrees of freedom of the space--do not depend on such features. Coarse geometry and coarse topology provide tools for measuring the large-scale properties of a space, and just as a metric or a topology contains information on the small-scale structure of a space, a coarse structure contains information on its large-scale properties.
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In mathematics, especially in category theory, a closed monoidal category (or a monoidal closed category) is a category that is both a monoidal category and a closed category in such a way that the structures are compatible. A classic example is the category of sets, Set, where the monoidal product of sets A and B is the usual cartesian product A \times B, and the internal Hom B^A is the set of functions from A to B. A non-cartesian example is the category of vector spaces, K-Vect, over a field K. Here the monoidal product is the usual tensor product of vector spaces, and the internal Hom is the vector space of linear maps from one vector space to another. The internal language of closed symmetric monoidal categories is linear logic and the type system is the linear type system. Many examples of closed monoidal categories are symmetric.
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In graph theory, the replacement product of two graphs is a graph product that can be used to reduce the degree of a graph while maintaining its connectivity. Suppose G is a d-regular graph and H is an e-regular graph with vertex set {0, …, d − 1}. Let R denote the replacement product of G and H. The vertex set of R is the Cartesian product V(G) × V(H). For each vertex u in V(G) and for each edge (i, j) in E(H), the vertex (u, i) is adjacent to (u, j) in R. Furthermore, for each edge (u, v) in E(G), if v is the ith neighbor of u and u is the jth neighbor of v, the vertex (u, i) is adjacent to (v, j) in R. If H is an e-regular graph, then R is an (e + 1)-regular graph.
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The axis-aligned minimum bounding box (or AABB) for a given point set is its minimum bounding box subject to the constraint that the edges of the box are parallel to the (Cartesian) coordinate axes. It is the Cartesian product of N intervals each of which is defined by the minimal and maximal value of the corresponding coordinate for the points in S. Axis- aligned minimal bounding boxes are used to an approximate location of an object in question and as a very simple descriptor of its shape. For example, in computational geometry and its applications when it is required to find intersections in the set of objects, the initial check is the intersections between their MBBs. Since it is usually a much less expensive operation than the check of the actual intersection (because it only requires comparisons of coordinates), it allows quickly excluding checks of the pairs that are far apart.
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H(3,3) drawn as a unit distance graph Hamming graphs are a special class of graphs named after Richard Hamming and used in several branches of mathematics and computer science. Let S be a set of q elements and d a positive integer. The Hamming graph H(d,q) has vertex set Sd, the set of ordered d-tuples of elements of S, or sequences of length d from S. Two vertices are adjacent if they differ in precisely one coordinate; that is, if their Hamming distance is one. The Hamming graph H(d,q) is, equivalently, the Cartesian product of d complete graphs Kq.. In some cases, Hamming graphs may be considered more generally as the Cartesian products of complete graphs that may be of varying sizes.. Unlike the Hamming graphs H(d,q), the graphs in this more general class are not necessarily distance-regular, but they continue to be regular and vertex-transitive.
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In fact these uses of AC are essential: in 1950 Kelley proved that Tychonoff's theorem implies the axiom of choice in ZF. Note that one formulation of AC is that the Cartesian product of a family of nonempty sets is nonempty; but since the empty set is most certainly compact, the proof cannot proceed along such straightforward lines. Thus Tychonoff's theorem joins several other basic theorems (e.g. that every vector space has a basis) in being equivalent to AC. On the other hand, the statement that every filter is contained in an ultrafilter does not imply AC. Indeed, it is not hard to see that it is equivalent to the Boolean prime ideal theorem (BPI), a well-known intermediate point between the axioms of Zermelo-Fraenkel set theory (ZF) and the ZF theory augmented by the axiom of choice (ZFC). A first glance at the second proof of Tychnoff may suggest that the proof uses no more than (BPI), in contradiction to the above.
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A given n-vertex graph G has an odd cycle transversal of size k, if and only if the Cartesian product of graphs G\square K_2 (a graph consisting of two copies of G, with corresponding vertices of each copy connected by the edges of a perfect matching) has a vertex cover of size n+k. The odd cycle transversal can be transformed into a vertex cover by including both copies of each vertex from the transversal and one copy of each remaining vertex, selected from the two copies according to which side of the bipartition contains it. In the other direction, a vertex cover of G\square K_2 can be transformed into an odd cycle transversal by keeping only the vertices for which both copies are in the cover. The vertices outside of the resulting transversal can be bipartitioned according to which copy of the vertex was used in the cover.
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The theorem can be stated simply as follows.Elementary Differential Equations and Boundary Value Problems (4th Edition), W.E. Boyce, R.C. Diprima, Wiley International, John Wiley & Sons, 1986, For the equation and initial value problem: : y' = F(x,y)\,,\quad y_0 = y(x_0) if F and ∂F/∂y are continuous in a closed rectangle :R=[x_0-a,x_0+a]\times [y_0-b,y_0+b] in the x-y plane, where a and b are real (symbolically: a, b ∈ ℝ) and × denotes the cartesian product, square brackets denote closed intervals, then there is an interval :I = [x_0-h,x_0+h] \subset [x_0-a,x_0+a] for some h ∈ ℝ where the solution to the above equation and initial value problem can be found. That is, there is a solution and it is unique. Since there is no restriction on F to be linear, this applies to non- linear equations that take the form F(x, y), and it can also be applied to systems of equations.
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Illustration of the axiom of choice, with each Si and xi represented as a jar and a colored marble, respectively family of sets indexed over the real numbers R; that is, there is a set Si for each real number i, with a small sample shown above. Each set contains at least one, and possibly infinitely many, elements. The axiom of choice allows us to arbitrarily select a single element from each set, forming a corresponding family of elements (xi) also indexed over the real numbers, with xi drawn from Si. In general, the collections may be indexed over any set I, not just R. In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that a Cartesian product of a collection of non-empty sets is non-empty. Informally put, the axiom of choice says that given any collection of bins, each containing at least one object, it is possible to make a selection of exactly one object from each bin, even if the collection is infinite.
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This addition of morphism turns Ab into a preadditive category, and because the direct sum of finitely many abelian groups yields a biproduct, we indeed have an additive category. In Ab, the notion of kernel in the category theory sense coincides with kernel in the algebraic sense, i.e. the categorical kernel of the morphism f : A → B is the subgroup K of A defined by K = {x ∈ A : f(x) = 0}, together with the inclusion homomorphism i : K → A. The same is true for cokernels; the cokernel of f is the quotient group C = B / f(A) together with the natural projection p : B → C. (Note a further crucial difference between Ab and Grp: in Grp it can happen that f(A) is not a normal subgroup of B, and that therefore the quotient group B / f(A) cannot be formed.) With these concrete descriptions of kernels and cokernels, it is quite easy to check that Ab is indeed an abelian category. The product in Ab is given by the product of groups, formed by taking the cartesian product of the underlying sets and performing the group operation componentwise.
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