Bands of E-Inversive Unipotent Semigroups

Abstract

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.

Some results on $$\mathcal {L}$$-commutative semigroups

“…This implies that S∕K S is a nil-semigroup. Also, K is the least rectangular band congruence on S. Consequently, S is a retract ideal extension of a completely simple semigroup by a nil-semigroup (see [6,Theorem 3.3]).…”

“…Hence It is now clear that S is a nil-extension of the left group K S by the commutative nilsemigroup S∕K S . Finally, K S is a retract ideal of S by [6,Theorem 3.3]. ◻…”