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In graph theory, a graph product is a binary operation on graphs. Specifically, it is an operation that takes two graphs G 1 and G 2 and produces a graph H with the following properties: The vertex set of H is the Cartesian product V ( G 1 ) × V ( G 2 ) , where V ( G 1 ) and V ( G 2 ) are the vertex sets of G 1 and G 2 , respectively.
Given two sets M and N endowed with monoid structure (or, in general, any finite number of monoids, M 1, ..., M k), their Cartesian product M × N, with the binary operation and identity element defined on corresponding coordinates, called the direct product, is also a monoid (respectively, M 1 × ⋅⋅⋅ × M k). [5] Fix a monoid M.
A two-sided identity (or just identity) is an element that is both a left and right identity. Semigroups with a two-sided identity are called monoids. A semigroup may have at most one two-sided identity. If a semigroup has a two-sided identity, then the two-sided identity is the only one-sided identity in the semigroup.
In category theory, a branch of mathematics, a monoid (or monoid object, or internal monoid, or algebra) (M, μ, η) in a monoidal category (C, ⊗, I) is an object M together with two morphisms. μ: M ⊗ M → M called multiplication, η: I → M called unit, such that the pentagon diagram. and the unitor diagram commute.
the tensor product of two objects A 1, ..., A n and B 1, ..., B m is the concatenation A 1, ..., A n, B 1, ..., B m of the two lists, and, similarly, the tensor product of two morphisms is given by the concatenation of lists. The identity object is the empty list. This operation Σ mapping category C to Σ(C) can be extended to a strict 2-monad ...
The Cartesian product of K 2 and a path graph is a ladder graph. The Cartesian product of two path graphs is a grid graph. The Cartesian product of n edges is a hypercube: =. Thus, the Cartesian product of two hypercube graphs is another hypercube: Q i Q j = Q i+j. The Cartesian product of two median graphs is another median graph.
In Cartesian closed categories, a "function of two variables" (a morphism f : X×Y → Z) can always be represented as a "function of one variable" (the morphism λf : X → Z Y). In computer science applications, this is known as currying ; it has led to the realization that simply-typed lambda calculus can be interpreted in any Cartesian ...
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. [1] 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 ...