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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.
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 ...
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.
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).
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"
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.
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[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.