Inverse Semigroup - Congruences On Inverse Semigroups

Congruences On Inverse Semigroups

Congruences are defined on inverse semigroups in exactly the same way as for any other semigroup: a congruence ρ is an equivalence relation which is compatible with semigroup multiplication, i.e.,

Of particular interest is the relation, defined on an inverse semigroup S by

there exists a with

It can be shown that σ is a congruence and, in fact, it is a group congruence, meaning that the factor semigroup S/σ is a group. In the set of all group congruences on a semigroup S, the minimal element (for the partial order defined by inclusion of sets) need not be the smallest element. In the specific case in which S is an inverse semigroup σ is the smallest congruence on S such that S/σ is a group, that is, if τ is any other congruence on S with S/τ a group, then σ is contained in τ. The congruence σ is called the minimum group congruence on S. The minimum group congruence can be used to give a characterisation of E-unitary inverse semigroups (see below).

A congruence ρ on an inverse semigroup S is called idempotent pure if