Monoid - Equational Presentation

Equational Presentation

Monoids may be given a presentation, much in the same way that groups can be specified by means of a group presentation. One does this by specifying a set of generators Σ, and a set of relations on the free monoid Σ∗. One does this by extending (finite) binary relations on Σ∗ to monoid congruences, and then constructing the quotient monoid, as above.

Given a binary relation R ⊂ Σ∗ × Σ∗, one defines its symmetric closure as R ∪ R−1. This can be extended to a symmetric relation E ⊂ Σ∗ × Σ∗ by defining x ~_E y 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 . Thus, for example,

is the equational presentation for the bicyclic monoid, and

is the plactic monoid of degree 2 (it has infinite order). Elements of this plactic monoid may be written as for integers i, j, k, as the relations show that ba commutes with both a and b.