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Presentation of a monoid

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Source updated March 4, 2025

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

A presentation should not be confused with a representation.

01Construction

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 ~_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 then is a monoid congruence.

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

$\langle p,q\,\vert \;pq=1\rangle$

is the equational presentation for the bicyclic monoid, and

$\langle a,b\,\vert \;aba=baa,bba=bab\rangle$

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

02Inverse monoids and semigroups

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

$(X;T)$

where

$(X\cup X^{-1})^{*}$

is the free monoid with involution on $X$ , and

$T\subseteq (X\cup X^{-1})^{*}\times (X\cup X^{-1})^{*}$

is a binary relation between words. We denote by $T^{\mathrm {e} }$ (respectively $T^{\mathrm {c} }$ ) the equivalence relation (respectively, the congruence) generated by T.

We use this pair of objects to define an inverse monoid

$\mathrm {Inv} ^{1}\langle X|T\rangle .$

Let $\rho _{X}$ be the Wagner congruence on $X$ , we define the inverse monoid

$\mathrm {Inv} ^{1}\langle X|T\rangle$

presented by $(X;T)$ as

$\mathrm {Inv} ^{1}\langle X|T\rangle =(X\cup X^{-1})^{*}/(T\cup \rho _{X})^{\mathrm {c} }.$

In the previous discussion, if we replace everywhere $({X\cup X^{-1}})^{*}$ with $({X\cup X^{-1}})^{+}$ we obtain a presentation (for an inverse semigroup) $(X;T)$ and an inverse semigroup $\mathrm {Inv} \langle X|T\rangle$ presented by $(X;T)$ .

A trivial but important example is the free inverse monoid (or free inverse semigroup) on $X$ , that is usually denoted by $\mathrm {FIM} (X)$ (respectively $\mathrm {FIS} (X)$ ) and is defined by

$\mathrm {FIM} (X)=\mathrm {Inv} ^{1}\langle X|\varnothing \rangle =({X\cup X^{-1}})^{*}/\rho _{X},$

$\mathrm {FIS} (X)=\mathrm {Inv} \langle X|\varnothing \rangle =({X\cup X^{-1}})^{+}/\rho _{X}.$

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This article is adapted from the Wikipedia article “Presentation of a monoid”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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