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Function f : [Z] 3 → [Z] 6 given by [k] 3 ↦ [3k] 6 is a semigroup homomorphism, since [3k ⋅ 3l] 6 = [9kl] 6 = [3kl] 6. However, f([1] 3) = [3] 6 ≠ [1] 6, so a monoid homomorphism is a semigroup homomorphism between monoids that maps the identity of the first monoid to the identity of the second monoid and the latter condition cannot be ...
M. Kilp, U. Knauer, A.V. Mikhalev, Monoids, Acts and Categories with Applications to Wreath Products and Graphs, De Gruyter Expositions in Mathematics vol. 29, Walter de Gruyter, 2000, ISBN 3-11-015248-7. Ronald V. Book and Friedrich Otto, String-rewriting Systems, Springer, 1993, ISBN 0-387-97965-4, chapter 7, "Algebraic Properties"
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.
[2] [3] The notation S 1 denotes a monoid obtained from S by adjoining an identity if necessary (S 1 = S for a monoid). [3] Similarly, every magma has at most one absorbing element, which in semigroup theory is called a zero. Analogous to the above construction, for every semigroup S, one can define S 0, a semigroup with 0 that embeds S.
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.
Monoidal categories can be seen as a generalization of these and other examples. Every ( small ) monoidal category may also be viewed as a " categorification " of an underlying monoid , namely the monoid whose elements are the isomorphism classes of the category's objects and whose binary operation is given by the category's tensor product.
A finite monoid is rational. A group is a rational monoid if and only if it is finite.; A finitely generated free monoid is rational. The monoid M4 generated by the set {0,e, a,b, x,y} subject to relations in which e is the identity, 0 is an absorbing element, each of a and b commutes with each of x and y and ax = bx, ay = by = bby, xx = xy = yx = yy = 0 is rational but not automatic.
In group theory, Cayley's theorem asserts that any group G is isomorphic to a subgroup of the symmetric group of G (regarded as a set), so that G is a permutation group.This theorem generalizes straightforwardly to monoids: any monoid M is a transformation monoid of its underlying set, via the action given by left (or right) multiplication.