Bands of E-Inversive Unipotent Semigroups

We study some special types of bands of E-inversive unipotent semigroups. It has been proved that in any R-semigroup S, which is a band of E-inversive unipotent semigroups, the set of its regular elements is a retract of S. Also, some characterizations of E-inversive rectangular bands of unipotent semigroups are given. This theorem extends nearly 40-old results from the theory of epigroups. In fact, a more general result is valid in some special subclass of the class of E-inversive semigroups; this result seems to be (partially) new for all epigroups.