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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 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.
The Hedetniemi conjecture, which gave a formula for the chromatic number of a tensor product, was disproved by Yaroslav Shitov . The tensor product of graphs equips the category of graphs and graph homomorphisms with the structure of a symmetric closed monoidal category. Let G 0 denote the underlying set of vertices of the graph G.
A binary relation between monoids M and N is a rational relation if the graph of the relation, regarded as a subset of M×N is a rational set in the product monoid. A function from M to N is a rational function if the graph of the function is a rational set. [4]
Let T, η, μ be a monad over a category C.The Kleisli category of C is the category C T whose objects and morphisms are given by = (), (,) = (,).That is, every morphism f: X → T Y in C (with codomain TY) can also be regarded as a morphism in C T (but with codomain Y).
A free group of finite rank n > 1 has an exponential growth rate of order 2n − 1. A few other related results are: The Nielsen–Schreier theorem: Every subgroup of a free group is free. Furthermore, if the free group F has rank n and the subgroup H has index e in F, then H is free of rank 1 + e(n–1).