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For example, monoids are semigroups with identity. In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids are semigroups with identity.
In algebra, a presentation of a monoid (or a presentation of a semigroup) is a description of a monoid (or a semigroup) in terms of a set Σ of generators and a set of relations on the free monoid Σ ∗ (or the free semigroup Σ +) generated by Σ. The monoid is then presented as the quotient of the free monoid (or the free semigroup) by these ...
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.
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 free group F S with free generating set S can be constructed as follows. S is a set of symbols, and we suppose for every s in S there is a corresponding "inverse" symbol, s −1, in a set S −1. Let T = S ∪ S −1, and define a word in S to be any written product of elements of T. That is, a word in S is an element of the monoid ...
Semigroup with two elements: there are five that are essentially different. A null semigroup on any nonempty set with a chosen zero, or a left/right zero semigroup on any set. The "flip-flop" monoid: a semigroup with three elements representing the three operations on a switch – set, reset, and do nothing. The set of positive integers with ...
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.