Presentation of a monoid

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

Contents

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 is then 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,\cdots,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^ib^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

(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}})^%2B 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,

or

\mathrm{FIS}(X)=\mathrm{Inv} \langle X | \varnothing\rangle=({X\cup X^{-1}})^%2B/\rho_X.

Notes

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

References