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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.
Many definitions and theorems about monoids can be generalised to small categories with more than one object. For example, a quotient of a category with one object is just a quotient monoid. Monoids, just like other algebraic structures, also form their own category, Mon, whose objects are monoids and whose morphisms are monoid homomorphisms. [8]
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 category of all directed graphs is Cartesian closed; this is a functor category as explained under functor category. In particular, the category of simplicial sets (which are functors X : Δ op → Set) is Cartesian closed. Even more generally, every elementary topos is Cartesian closed.
A 16-year-old New Mexico boy was arrested Saturday morning after he allegedly gunned down his parents and teenage siblings and then drunkenly called 911 to confess to the quadruple homicide ...
The two-sided Krohn–Rhodes theorem alternatively characterizes finite aperiodic monoids as divisors of iterated block products of copies of the two-element semilattice. See also [ edit ]
The College Football Playoff got underway Friday but the main course is spread out through Saturday. Three first-round games will be played across three separate campus sites from State College ...