Onesided Inverses for Semigroups

In any semigroup S, we say that elements a and b are left inverses of each other if a = aba, b = bab and a L b, in which case we write a γ b. Right inverses are defined dually with the notation δ. Set τ = δγ. We study the classes of semigroups in which τ has some of the usual properties of a relation. We also consider properties of (maximal) completely simple subsemigroups of S. In terms of the above concepts, we characterize E–solid, central, (almost) L–unipotent and locally (almost) L–unipotent semigroups in many ways. We define these notions for arbitrary semigroups by extending their definitions from regular semigroups.