A generalized Clifford theorem of semigroups

A U-abundant semigroup S in which every (H) over tilde -class of S contains an element in the set of projections U of S is said to be a U-superabundant semigroup. This is an analogue of regular semigroups which are unions of groups and an analogue of abundant semigroups which are superabundant. In 1941, Clifford proved that a semigroup is a union of groups if and only if it is a semilattice of completely simple semigroups. Several years later, Fountain generalized this result to the class of superabundant semigroups. In this paper, we extend their work to U-superabundant semigroups.