Standard regular semigroups

Abstract

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 Y-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 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-> 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 b Q H b -i rb , (c) g(c, d)eH (cd) -i cd . 2(a) hB eyh y = h*e y for feeJ and i/e. (b) if j e H z and i e 7 Z , JJ5< = j (c) g(i/, 2) = e yz for y,2e7. 3(a) c d = cd C g{c , d) (xC z = z-'*x*z for x,zeH) (b) #(, bg(b, c) = g(ab, c)*(g(a, b) c ). Let (, /, J, F, , /3, ^r) denote {(, , i): e F, i e I aa ~, j e J a ~a } under the multiplication (4) (i f a> 3){w, , v) = (io (6)(6) -i, 6, flf(, b)*jB w b *v) .