Extensions of left regular bands by $${\mathcal {R}}$$-unipotent semigroups

In this paper we describe $${\mathcal {R}}$$ -unipotent semigroups being regular extensions of a left regular band by an $$\mathcal {R}$$ -unipotent semigroup T as certain subsemigroups of a wreath product of a left regular band by T. We obtain Szendrei’s result that each E-unitary $${\mathcal {R}}$$ -unipotent semigroup is embeddable into a semidirect product of a left regular band by a group. Further, specialising the first author’s notion of $$\lambda $$ -semidirect product of a semigroup by a locally $${\mathcal {R}}$$ -unipotent semigroup, we provide an answer to an open question raised by the authors in [Extensions and covers for semigroups whose idempotents form a left regular band, Semigroup Forum 81 (2010), 51-70].