$J$-compatible orthodox semigroups
Miyuki Yamada · Proceedings of the Japan Academy Series A Mathematical Sciences · 1979
A semigroup S is said to be J-compatible if Green's J-relation is a congruence on S. In this paper, we shll study the structure of J- compatible orthodox semigroups.1. Basic properties.LetS bea regular semigroup.Throughout this paper, the J-relation and the D-relation on S will be denoted by fls and 2s respectively.Further, the congruence generated by will be denoted by fl*[].Let ]s be the least semilattice congruence on S.Then, it has been show by Hall [2] that _q)* =]s.Further, it is easily see that .In fact, if (a, b) e then there exists c e S such that a.csb, where _s and s denote the L-relation and the R-relation on S respectively.Hence, Sa=Sc and cS-bS.Accord- ingly, SaS=ScS--SbS.Therefore, (a, b)e fls.Since ]s is the least semilattice congruence on S,S is a semilattice F of the ]s-classes (S"e F} (in this case, F S/]) and each S is semilattice-indecompos- able.Ifacgsb, thenSaS=SbS.Hence