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.

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

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

• 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"