Semilattice of Bisimple Regular Semigroups

The main purpose of this paper is to show that a regular semigroup S is a semilattice of bisimple semigroups if and only if it is a band of bisimple semigroups and that this holds if and only if 3) is a congruence on S. It is also shown that a quasiregular semigroup 5 which is a rectangular band of bisimple semigroups is itself bisimple. In [3, Theorem 4.4] it was shown that a semigroup S is a semilattice of simple semigroups if and only if it is a union of simple semigroups. The purpose of this paper is to obtain corresponding results for a semigroup which is a semilattice of bisimple regular semigroups. Unfortunately, a semilattice of bisimple semigroups need not be a union of bisimple semigroups as illustrated by a simple w-semigroup constructed by Munn [S]. However, we get some equivalent conditions for such semigroups. In particular we show that a regular semigroup is a semilattice of bisimple semigroups if and only if it is a band of bisimple semigroups. 1. Equivalent conditions. In this section we consider a set of equivalent conditions for a semigroup 5 to be a semilattice of bisimple semigroups. We adopt the terminology and notation of [2]. Lemma 1.1. Let S be a semilattice ß of semigroups Sa and let D be a Si-class of S. Then, either Sai^D = □ or DQSa. Proof. Suppose Sar\D?£\Zi. Let aaESa(~\D. If bßESß and aaS)bß, then there exists cyESy (/y£ß) such that aa6icy and cy£bß. Also aa6iCy implies that either aa=cy in which case y= a, or there exist X\ES\, y^ESp (X, n in ß) such that aax\=cy and cyyß = aa. However, since S is a semilattice of the semigroups Sa, aax\ESa\ and cyy„ ESyt¡. It follows that7=aX and a=yn and so y^a and a^y. Thus, in either case, 7 = a. Likewise, y=ß. Therefore, a = ß and D<^Sa. Lemma 1.2. Let S be a semigroup. If abS)ba for all a, b in S, then 3) is a congruence on S. Proof. Let a£>b. Then there exists xES such that a£x and xtRb. Since £ is a right congruence and (R is a left congruence, we have Received by the editors July 3, 1970. AMS 1969 subject classifications. Primary 2092, 2093.