By an inverse transversal of a regular semigroup S we mean an inverse subsemigroup that contains a single inverse of every element of S. A certain multiplicative property (which in the case of a band is equivalent to normality) is imposed on an inverse transversal and a complete description of the structure of S is obtained.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0308210500032522

Regular semigroups with a multiplicative inverse transversal

Proper extensions that are “injective on ℒ-related idempotents” 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 semigroups, are described as certain subalgebras of a λ-semidirect product of a left regular band by an ℛ-unipotent or by a glrac semigroup, 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 semilattice

https://www-users.york.ac.uk/~varg1/properextensions.pdf

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

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0308210500016140

Constance C. Edwards

Papers

The greatest idempotent separating congruence on an ℒ-unipotent semigroup