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 $R ∪ R^{&minus;1}$ 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 $(u,v) ∈ R ∪ R^{&minus;1}$. 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 $$a^ib^j(ba)^k$$ 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 $$({X\cup  X^{-1}})^+$$ 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
 * $$\mathrm{FIS}(X)=\mathrm{Inv} \langle X | \varnothing\rangle=({X\cup X^{-1}})^+/\rho_X.$$