Green's relations and quasi-ideals
Kenneth M. Kapp · Czechoslovak Mathematical Journal · 1969
In generalizing Croisot's Theory of Decompositions [2] we came upon the subclass of regular absorbent semigroups.It was shown ([2] Corollary (3.6)) that a semigroup belonged to this class precisely when it was the mutually annihilating sum of subsemigroups each of which was a completely 0-simple semigroup when considered sepa rately.OTTO STEINFELD has also found equivalent conditions [7] for such a decom position.His results were in part a culmination of his investigations of quasi-ideals in semigroups ([4], [5], [6]).It is our purpose now to reconcile both these approaches and to examine quasi-ideals from the standpoint of Green's relations.We give an interesting characterization of absorbency in terms of a partial order on Green's equivalence classes (1.6).We then show that for an absorbent semigroup with 0 each c^-class union 0 is a quasi-ideal (1.8) (indeed, these are the only 0minimal quasi-ideals (3.6)).For regular or commutative semigroups we show that the converse is true.It is an open question as to whether the converse is always true.Finally we give a direct proof of two theorems and corollaries closely related to one of Steinfeld's, using techniques involving Green's relations.