 We denote by formula_9 (respectively formula_10) the equivalence relation (respectively, the congruence) generated by "T".We use this pair of objects to define an inverse monoid

Let formula_12 be the Wagner congruence on formula_7, we define the inverse monoid

"presented" by formula_5 as

In the previous discussion, if we replace everywhere formula_17 with formula_18 we obtain a presentation (for an inverse semigroup) formula_5 and an inverse semigroup formula_20 presented by formula_5.A trivial but important example is the free inverse monoid (or free inverse semigroup) on formula_7, that is usually denoted by formula_23 (respectively formula_24) and is defined by

or