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The monoids from AND and OR are also idempotent while those from XOR and XNOR are not. The set of natural numbers N = {0, 1, 2, ...} is a commutative monoid under addition (identity element 0) or multiplication (identity element 1). A submonoid of N under addition is called a numerical monoid.
Any category with finite products can be regarded as monoidal with the product as the monoidal product and the terminal object as the unit. Such a category is sometimes called a cartesian monoidal category. For example: Set, the category of sets with the Cartesian product, any particular one-element set serving as the unit.
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.
In category theory, monoidal functors are functors between monoidal categories which preserve the monoidal structure. More specifically, a monoidal functor between two monoidal categories consists of a functor between the categories, along with two coherence maps—a natural transformation and a morphism that preserve monoidal multiplication and unit, respectively.
First, one takes the symmetric closure R ∪ R −1 of R. This is then extended to a symmetric relation E ⊂ Σ ∗ × Σ ∗ by defining x ~ E y if and only if x = sut and y = svt for some strings u, v, s, t ∈ Σ ∗ with (u,v) ∈ R ∪ R −1. Finally, one takes the reflexive and transitive closure of E, which then is a monoid congruence.
An optimal five-vertex dominating set in the product of two stars, K 1,4 K 1,4. Examples such as this one show that, for some graph products, Vizing's conjecture can be far from tight. A 4-cycle C 4 has domination number two: any single vertex only dominates itself and its two neighbors, but any pair of vertices dominates the whole graph.
History monoids were first presented by M.W. Shields. [1] History monoids are isomorphic to trace monoids (free partially commutative monoids) and to the monoid of dependency graphs. As such, they are free objects and are universal. The history monoid is a type of semi-abelian categorical product in the category of monoids.
For the language (+), the minimal automaton has 4 states and the syntactic monoid has 15 elements. [ 10 ] The bicyclic monoid is the syntactic monoid of the Dyck language (the language of balanced sets of parentheses).