Generalised domain and E-inverse semigroups

Abstract

A generalised D-semigroup is here defined to be a left E-semiabundant semigroup S in which the $\overline{\mathcal R}_E$-class of every $x\in S$ contains a unique element D(x) of E, made into a unary semigroup. Two-sided versions are defined in the obvious way in terms of $\overline{\mathcal R}_E$ and $\overline{\mathcal L}_E$. The resulting class of unary (bi-unary) semigroups is shown to be a finitely based variety, properly containing the variety of D-semigroups (defined in an order-theoretic way in Communications in Algebra, 3979–4007, 2014). Important subclasses associated with the regularity and abundance properties are considered. The full transformation semigroup $T_X$ can be made into a generalised D-semigroup in many natural ways, and an embedding theorem is given. A generalisation of inverse semigroups in which inverses are defined relative to a set of idempotents arises as a special case, and a finite equational axiomatisation of the resulting unary semigroups is given.