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.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.