 Thus, for example,
is the equational presentation for the bicyclic monoid, and

is the plactic monoid of degree 2 (it has infinite order).Elements of this plactic monoid may be written as formula_4 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
where

formula_6

is the free monoid with involution on formula_7, and

is a binary relation between words.