D*-simple inverse semigroups

Abstract

Let ~ s be the Green's D-relation on a semigroup S, and ~s* the congruence on S generated by ~ s , that is, the least congruence (on S) containing ~ s I f ~s* is the universal congruence, that is, if ~s* satisfies a~*sb for all a, b E S, then S is said to be D*-simple. According to Hall [1], the least semilattice congruence ~s on a regular semigroup S coincides with 2s*; that is, ~?s = ~s*. From this fact, it follows from Petrich [6] that the ~Us-relation , U s on a regular semigroup S in the sense of [6] also coincides with ~s*. We shall call ~s* the D*-relation on S. Now, let S be a regular semigroup. There exist a semilattice A and a regular semigroup Sa for each A ~ A such that S is a semilattice A of the regular semigroups S a and each S a is an ~?s-class. In this case, each Sa is semilattice indecom: posable (S-indecomposable) and hence S a is D*-simple. Tha t is, a regular [orthodox, inverse] semigroup S is a semilattice of D*-simple regular [orthodox , inverse] semigroups. Conversely, let A be a semilattice and Sa an orthodox semigroup for each A ~ A. Let S(o) be a semilattice composition of (Sa:. A t A}; that is, S(o) is a semigr0up such that