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 g b . Right inverses are defined dually with the notation d . Set t = bg . We study the classes of semigroups in which t 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.