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  2. Monoid - Wikipedia

    en.wikipedia.org/wiki/Monoid

    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]

  3. Presentation of a monoid - Wikipedia

    en.wikipedia.org/wiki/Presentation_of_a_monoid

    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"

  4. Monoid (category theory) - Wikipedia

    en.wikipedia.org/wiki/Monoid_(category_theory)

    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.

  5. Graph product - Wikipedia

    en.wikipedia.org/wiki/Graph_product

    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.

  6. Category:Graph products - Wikipedia

    en.wikipedia.org/wiki/Category:Graph_products

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  7. Cartesian closed category - Wikipedia

    en.wikipedia.org/wiki/Cartesian_closed_category

    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.

  8. Product (category theory) - Wikipedia

    en.wikipedia.org/wiki/Product_(category_theory)

    In the category of graphs, the product is the tensor product of graphs. In the category of relations, the product is given by the disjoint union. (This may come as a bit of a surprise given that the category of sets is a subcategory of the category of relations.) In the category of algebraic varieties, the product is given by the Segre embedding.

  9. Three utilities problem - Wikipedia

    en.wikipedia.org/wiki/Three_utilities_problem

    , is a graph with six vertices and nine edges, often referred to as the utility graph in reference to the problem. [1] It has also been called the Thomsen graph after 19th-century chemist Julius Thomsen. It is a well-covered graph, the smallest triangle-free cubic graph, and the smallest non-planar minimally rigid graph.