A term used as an abbreviation for the phrase "semi-group with identity" . Thus, a monoid is a set with an associative binary operation, usually called multiplication, in which there is an element such that for any . The element is called the identity (or unit) and is usually denoted by . In any monoid there is exactly one identity. If the operation given on the monoid is commutative, it is often called addition and the identity is called the zero and is denoted by .
Examples of monoids. 1) The set of all mappings of an arbitrary set into itself is a monoid relative to the operation of successive application (composition) of mappings. The identity mapping is the identity. 2) The set of endomorphisms of a universal algebra is a monoid relative to composition; the identity is the identity endomorphism. 3) Every group is a monoid.
Every semi-group without an identity can be imbedded in a monoid. For this it suffices to take a symbol not in and give a multiplication on the set as follows: , for any , and on elements from the operation is as before. Every monoid can be represented as the monoid of all endomorphisms of some universal algebra.
An arbitrary monoid can also be considered as a category with one object. This allows one to associate with a monoid its dual (opposite, adjoint) monoid . The elements of both monoids coincide, but the product of and in is put equal to the product in .
The development of the theory of monoids and adjoint functors has shown the utility of the definition of a monoid in so-called monoidal categories. Suppose given a category equipped with a bifunctor , an object and natural isomorphisms
satisfying coherence conditions. An object is called a monoid in the category if there are morphisms and such that the following diagrams are commutative:
If is taken to be the category of sets, the Cartesian product, a one-point set, and the isomorphisms , and are chosen in the natural way (, ), then the second definition of a monoid turns out to be equivalent to the original definition.
|||A.H. Clifford, G.B. Preston, "Algebraic theory of semi-groups" , 1–2 , Amer. Math. Soc. (1961–1967)|
|||S. MacLane, "Categories for the working mathematician" , Springer (1971)|
For monoidal categories, and particularly the coherence conditions that the isomorphisms , must satisfy cf. , Chapt. 7, Sects. 1-2.
Monoid. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Monoid&oldid=29746