Editing Monoid (section)

== Equational presentation ==
{{main|Presentation of a monoid}}

Monoids may be given a ''presentation'', much in the same way that groups can be specified by means of a [[group presentation]]. One does this by specifying a set of generators {{math|Σ}}, and a set of relations on the [[free monoid]] {{math|Σ<sup>∗</sup>}}. One does this by extending (finite) [[binary relation]]s on {{math|Σ<sup>∗</sup>}} to monoid congruences, and then constructing the quotient monoid, as above.

Given a binary relation {{math|''R'' ⊂ Σ<sup>∗</sup> × Σ<sup>∗</sup>}}, one defines its symmetric closure as {{math|''R'' ∪ ''R''<sup>−1</sup>}}. This can be extended to a symmetric relation {{math|''E'' ⊂ Σ<sup>∗</sup> × Σ<sup>∗</sup>}} by defining {{math|''x'' ~<sub>''E''</sub> ''y''}} if and only if {{math|1=''x'' = ''sut''}} and {{math|1=''y'' = ''svt''}} for some strings {{math|''u'', ''v'', ''s'', ''t'' ∈ Σ<sup>∗</sup>}} with {{math|(''u'',''v'') ∈ ''R'' ∪ ''R''<sup>−1</sup>}}. Finally, one takes the reflexive and transitive closure of {{math|''E''}}, which is then a monoid congruence.

In the typical situation, the relation {{math|''R''}} is simply given as a set of equations, so that {{math|1=''R'' = {{mset|1=''u''<sub>1</sub> = ''v''<sub>1</sub>, ..., ''u''<sub>''n''</sub> = ''v''<sub>''n''</sub>}}}}. Thus, for example,
: <math>\langle p,q\,\vert\; pq=1\rangle</math>
is the equational presentation for the [[bicyclic monoid]], and
: <math>\langle a,b \,\vert\; aba=baa, bba=bab\rangle</math>
is the [[plactic monoid]] of degree {{math|2}} (it has infinite order). Elements of this plactic monoid may be written as <math>a^ib^j(ba)^k</math> for integers {{math|''i''}}, {{math|''j''}}, {{math|''k''}}, as the relations show that {{math|''ba''}} commutes with both {{math|''a''}} and {{math|''b''}}.