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 (k-turnregular semigroup) gives a description for E-inversive semigroups (for all natural numbers k). The second of these two (almost k-turninverse) does not yet have a description, although it has description in the class of completely regular semigroups - there they are precisely Rees matrix semigroups without zero.
All the material with more details is available in English and Estonian in www.math.ut.ee/~integral/inverse.