Monoids

Abbreviation: Mon

Definition

A monoid is a structure M = (M,·,e), where · is an infix binary operation, called the monoid product, and e is a constant (nullary operation), called the identity element , such that
· is associative:   (x·y)·z = x·(y·z),
e is an identity for ·:   e·x = x  and  x·e = x.

Morphisms

Let M and N be monoids. A morphism from M to N is a function h : M→N that is a homomorphism: h(x·y) = h(x)·h(y)  and  h(e) = e.

Some results

Examples

(X^X,o,id_X), the collection of functions on a sets X, with composition, and identity map.

(M(V)_n,·,I_n), the collection of n×n matrices over a vector space V, with matrix multiplication and identity matrix.

(Σ^*,·,λ), the collection of strings over a set Σ, with concatenation and the empty string. This is the free monoid generated by Σ.

Properties

Finite members

Subclasses

Superclasses