Presentation of a monoid

In algebra, a presentation of a monoid (or semigroup) is a description of a monoid (or semigroup) in terms of a set Σ of generators and a set of relations on the free monoid Σ^∗ (or free semigroup Σ⁺) generated by Σ. The monoid is then presented as the quotient of the free monoid by these relations. This is an analogue of a group presentation in group theory.

As a mathematical structure, a monoid presentation is identical to a string rewriting system (also known as semi-Thue system). Every monoid may be presented by a semi-Thue system (possibly over an infinite alphabet).

A presentation should not be confused with a representation.

The relations are given as a (finite) binary relation R on Σ^∗. To form the quotient monoid, these relations are extended to monoid congruences as follows.

First, one takes the symmetric closure R ∪ R⁻¹ of R. This is then extended to a symmetric relation E ⊂ Σ^∗ × Σ^∗ by defining x ~_Ey if and only if x = sut and y = svt for some strings u, v, s, t ∈ Σ^∗ with (u,v) ∈ R ∪ R⁻¹. Finally, one takes the reflexive and transitive closure of E, which is then a monoid congruence.

In the typical situation, the relation R is simply given as a set of equations, so that ${\displaystyle R=\{u_{1}=v_{1},\cdots ,u_{n}=v_{n}\}}$. Thus, for example,

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