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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.

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 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. 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"

• Semigroup theory