We give a structure theorem for
a class of regular semigroups. Let S be a regular semigroup, let T denote
the union of the maximal subgroups of S, and let E(T) denote the set of
idempotents of T. Assume T is a semigroup (equivalently, T is a semilattice Y of
completely simple semigroups (Ty: y ∈ Y )). If Y has a greatest element and
e,f,g ∈ E(T), e ≥ f, and e ≥ g imply fg = gf, we term S a standard regular
semigroup. The structure of S is given modulo right groups and an inverse
semigroup V in which every subgroup is a single element by means of an
explict multiplication. We specialize the structure theorem to orthodox,
ℒ-unipotent, and inverse semigroups, and to a class of semigroups with Y an
ωY -semilattice.
