Standard regular semigroups
R. J. Warne · Pacific Journal of Mathematics · 1976
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 (equivalents, T is a semilattice Y of completely simple semigroups (T y : y e Y)).If Y has a greatest element and e, f, ge E{T), e > /, and e > g imply fg -gf 9 we term S a standard regular gemigroup.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, -^-unipotent, and inverse semigroups, and to a class of semigroups with Y an ωY-semilattice.Finally, we show that S is a regular extension of T by V in the sense of Yamada [19].Let us first state the structure theorem.Let Y be a semilattice with greatest element.Let V be an inverse semigroup with semilattice of idempotents Y such that each subgroup of V consists of a single element.Let (/, o) be a standard regular semilattice Y of left zero semigroups (I y : y e Y).Let (J, *) be a standard regular semilattice Y of right groups (J y :y e Y).Suppose I y Π J v -{e y }, a single idempotent element, and e*e z = e y oe z = e yz for all y, z e Y. Let H y denote the maximal subgroup of J y containing e y .Let i-*Bi be a homomorphism of (/, °) into P(J), the semigroup of right translations of (J, *); let b->β b be a mapping of V into End(J, *), the semigroup of endomorphism of (J, *), and let g be a mapping of V x V into H = U (H y : p7),a semilattice Y of groups (H y : yeY) (with respect to the multiplication * in J) such that l(a) jB t e H yz for iel y and j G J z , (b) J r β b Q H b -i rb , (c) g(c, d)eH (cd) -i cd .2(a) hB eyhβ y = h*e y for feeJ and i/eΓ.(b) if j e H z and i e 7 Z , JJ5 3){w, δ, v) = (ioβ (β6)(β6) -i, α6, flf(α, b)*jB w β b *v) .We show (Theorem 3.14) that (Y, /, J, F, B, β, g) is a standard regular semigroup, and, conversely, every standard regular semigroup is isomorphic to some (F, /, J, V, B, β, g).