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Its unit element is the class of the ordinary 2-sphere. Furthermore, if a denotes the class of the torus, and b denotes the class of the projective plane, then every element c of the monoid has a unique expression in the form c = na + mb where n is a positive integer and m = 0, 1, or 2. We have 3b = a + b.
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.
John M. Howie, Fundamentals of Semigroup Theory (1995), Clarendon Press, Oxford ISBN 0-19-851194-9 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 .
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.
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.
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Monads are to monoids as comonads are to comonoids. Every set is a comonoid in a unique way, so comonoids are less familiar in abstract algebra than monoids; however, comonoids in the category of vector spaces with its usual tensor product are important and widely studied under the name of coalgebras.