The basic idea is the definition of *k*-inverse elements:

We study three main definitions related to *k*-inverse elements.
All these definitions are given so that for *k*=1 it is precisely
the definition of inverse semigroup.

The first of them (__ k-inverse semigroup__) gives a description for
semilattices (instead of generalization), and two weaker variants of it give
descriptions for regular semigroups.

The second definition (__weakly k-inverse semigroup__) which seemingly
generalizes the first, is also precisely a semilattice.

The third definition (__ k-turninverse semigroup__) gives a description
for groups. Two weaker variants of the main definition are different. The weakest
one (

All the material with more details is available in English and Estonian in www.math.ut.ee/~integral/inverse.