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

Abstract

It is well known that the semigroup \(\mathcal{B}(S)\) 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 \(\mathcal{B}(S)\) 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 \(\mathcal{B}(S)\) is in each of the seven nontrivial subvarieties of normal bands. We also show that the set \(\mathcal{B}_{m}(S)\) of all minimal bi-ideal elements of \(S\) forms a rectangular band and that \(\mathcal{B}_{m}(S)\) is a bi-ideal of the semigroup~\(\mathcal{B(S)}\).