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For example, monoids are semigroups with identity. In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids are semigroups with identity.
A monoid object in the category of monoids (with the direct product of monoids) is just a commutative monoid. This follows easily from the Eckmann–Hilton argument. A monoid object in the category of complete join-semilattices Sup (with the monoidal structure induced by the Cartesian product) is a unital quantale.
is closed under division. is closed under taking finite Cartesian products. The first condition is equivalent to stating that V is closed under taking subsemigroups and under taking quotients. The second property implies that the empty product—that is, the trivial semigroup of one element—belongs to each variety.
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.
Congruence classes and factor monoids are the objects of study in string rewriting systems. A nuclear congruence on S is one that is the kernel of an endomorphism of S. [6] A semigroup S satisfies the maximal condition on congruences if any family of congruences on S, ordered by inclusion, has a maximal element.
Mac Lane's planarity criterion uses this idea to characterize the planar graphs in terms of the cycle bases: a finite undirected graph is planar if and only if it has a sparse cycle basis or 2-basis, [3] a basis in which each edge of the graph participates in at most two basis cycles. In a planar graph, the cycle basis formed by the set of ...
In mathematics, the term cycle decomposition can mean: Cycle decomposition (graph theory), a partitioning of the vertices of a graph into subsets, such that the vertices in each subset lie on a cycle; Cycle decomposition (group theory), a useful convention for expressing a permutation in terms of its constituent cycles
A join-semilattice with zero is a refinement monoid if and only if it is distributive.. Any abelian group is a refinement monoid.. The positive cone G + of a partially ordered abelian group G is a refinement monoid if and only if G is an interpolation group, the latter meaning that for any elements a 0, a 1, b 0, b 1 of G such that a i ≤ b j for all i, j<2, there exists an element x of G ...