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Extensions and covers for semigroups whose idempotents form a left regular band

Branco, Mário J. J.; Gould, Victoria; Gomes, Gracinda M. S.

Semigroup Forum, 81 (2010), 51-70

Proper extensions that are “injective on L-related idem-
potents” of R-unipotent semigroups, and much more generally of the class of generalised left restriction semigroups possessing the ample and congruence conditions, referred to here as glrac semi-groups, are described as certain ubalgebras of a -semidirect product of a left regular band by an R-unipotent or by a glrac semi-group, respectively. An example of such is the generalized Szendrei
expansion. As a consequence of our embedding, we are able to give a structure theorem for proper left restriction semigroups. Further, we show that any glrac semigroup S has a proper cover that is a semidirect product of a left regular band by a monoid, and if S is left restriction, the left regular band may be taken to be a semi-lattice.