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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 .
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.
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.
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.
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
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Amtrak reported an adjusted operating loss of $705 million for the 12 months ending Sept. 30, down 9% over 2023. Amtrak's prior record was 32.3 million passengers in 2019, which was up 45% from ...
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.