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

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 $\mathcal{B(S)}$.