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).

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 [math]\displaystyle{ R=\{u_1=v_1,\ldots,u_n=v_n\} }[/math]. Thus, for example,

[math]\displaystyle{ \langle p,q\,\vert\; pq=1\rangle }[/math]

is the equational presentation for the bicyclic monoid, and

[math]\displaystyle{ \langle a,b \,\vert\; aba=baa, bba=bab\rangle }[/math]

is the plactic monoid of degree 2 (it has infinite order). Elements of this plactic monoid may be written as [math]\displaystyle{ a^ib^j(ba)^k }[/math] 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

[math]\displaystyle{ (X;T) }[/math]

where

[math]\displaystyle{ (X\cup X^{-1})^* }[/math]

is the free monoid with involution on [math]\displaystyle{ X }[/math], and

[math]\displaystyle{ T\subseteq (X\cup X^{-1})^*\times (X\cup X^{-1})^* }[/math]

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

We use this pair of objects to define an inverse monoid

[math]\displaystyle{ \mathrm{Inv}^1 \langle X | T\rangle. }[/math]

Let [math]\displaystyle{ \rho_X }[/math] be the Wagner congruence on [math]\displaystyle{ X }[/math], we define the inverse monoid

[math]\displaystyle{ \mathrm{Inv}^1 \langle X | T\rangle }[/math]

presented by [math]\displaystyle{ (X;T) }[/math] as

[math]\displaystyle{ \mathrm{Inv}^1 \langle X | T\rangle=(X\cup X^{-1})^*/(T\cup\rho_X)^{\mathrm{c}}. }[/math]

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

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

[math]\displaystyle{ \mathrm{FIM}(X)=\mathrm{Inv}^1 \langle X | \varnothing\rangle=({X\cup X^{-1}})^*/\rho_X, }[/math]

or

[math]\displaystyle{ \mathrm{FIS}(X)=\mathrm{Inv} \langle X | \varnothing\rangle=({X\cup X^{-1}})^+/\rho_X. }[/math]

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"