Geodesic
On the $le$-semigroups whose semigroup of~bi-ideal elements is a normal band
Algebra and discrete mathematics, Tome 20 (2015) no. 2, pp. 171-181.

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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 $\mathcal{B(S)}$.
Keywords: bi-ideal elements, duo; intra-regular, lattice-ordered semigroup, locally testable, normal band, regular. 
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A. K. Bhuniya; M. Kumbhakar. On the $le$-semigroups whose semigroup of~bi-ideal elements is a normal band. Algebra and discrete mathematics, Tome 20 (2015) no. 2, pp. 171-181. http://geodesic.mathdoc.fr/item/ADM_2015_20_2_a0/

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