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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.
Given two sets M and N endowed with monoid structure (or, in general, any finite number of monoids, M 1, ..., M k), their Cartesian product M × N, with the binary operation and identity element defined on corresponding coordinates, called the direct product, is also a monoid (respectively, M 1 × ⋅⋅⋅ × M k). [5] Fix a monoid M.
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.
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"
The regular languages over an alphabet A are the closure of the finite subsets of A*, the free monoid over A, under union, product, and generation of submonoid. [6] For the case of concurrent computation, that is, systems with locks, mutexes or thread joins, the computation can be described with history monoids and trace monoids. Roughly ...
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.
Trace monoids are commonly used to model concurrent computation, forming the foundation for process calculi. They are the object of study in trace theory . The utility of trace monoids comes from the fact that they are isomorphic to the monoid of dependency graphs ; thus allowing algebraic techniques to be applied to graphs , and vice versa.