Presentation of a monoid

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).^[1]

A presentation should not be confused with a representation.

Construction

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 ${\displaystyle R=\{u_{1}=v_{1},\ldots ,u_{n}=v_{n}\}}$. Thus, for example,

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

is the equational presentation for the bicyclic monoid, and

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

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

Inverse monoids and semigroups

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

${\displaystyle (X;T)}$

where

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

is the free monoid with involution on ${\displaystyle X}$, and

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

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

We use this pair of objects to define an inverse monoid

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

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

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

presented by ${\displaystyle (X;T)}$ as

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

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

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

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

or

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

Notes

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

References

• 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"