Presentation of a monoid

From Wikipedia, the free encyclopedia
Jump to navigation Jump to search

In algebra, a presentation of a monoid (respectively, a presentation of a semigroup) is a description of a monoid (respectively, a semigroup) in terms of a set Σ of generators and a set of relations on the free monoid Σ (respectively, the free semigroup Σ+) generated by Σ. The monoid is then presented as the quotient of the free monoid (respectively, the free semigroup) 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 a semi-Thue system). Every monoid may be presented by a semi-Thue system (possibly over an infinite alphabet).[1]

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 RR−1 of R. This is then 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) ∈ RR−1. Finally, one takes the reflexive and transitive closure of E, which then is 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.

Inverse monoids and semigroups[edit]

Presentations of inverse monoids and semigroups can be defined in a similar way using a pair


is the free monoid with involution on , and

is a binary relation between words. We denote by (respectively ) the equivalence relation (respectively, the congruence) generated by T.

We use this pair of objects to define an inverse monoid

Let be the Wagner congruence on , we define the inverse monoid

presented by as

In the previous discussion, if we replace everywhere with we obtain a presentation (for an inverse semigroup) and an inverse semigroup presented by .

A trivial but important example is the free inverse monoid (or free inverse semigroup) on , that is usually denoted by (respectively ) and is defined by



  1. ^ Book and Otto, Theorem 7.1.7, p. 149


  • John M. Howie, Fundamentals of Semigroup Theory (1995), Clarendon Press, Oxford ISBN 0-19-851194-9
  • M. Kilp, U. Knauer, A.V. Mikhalev, Monoids, Acts and Categories with Applications to Wreath Products and Graphs, De Gruyter Expositions in Mathematics vol. 29, Walter de Gruyter, 2000, ISBN 3-11-015248-7.
  • Ronald V. Book and Friedrich Otto, String-rewriting Systems, Springer, 1993, ISBN 0-387-97965-4, chapter 7, "Algebraic Properties"