On the le-semigroups whose semigroup of bi-ideal elements is a normal band

Abstract

It is well known that the semigroup B(S) of all bi-ideal elements of an le-semigroup S is a band if and only if S is both regular and intra-regular. Here we show that B(S) is a band if and only if it is a normal band and give a complete characterization of the le-semigroups S for which the associated semigroup B(S) is in each of the seven nontrivial subvarieties of normal bands. We also show that the set Bm(S) of all minimal bi-ideal elements of S forms a rectangular band and that Bm(S) is a bi-ideal of the semigroup B(S).