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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.
Type
Journal Article
Type of thesis
Series
Citation
Stokes, T. E. (2018). Generalised domain and E-inverse semigroups. Semigroup Forum, Online First. https://doi.org/10.1007/s00233-018-9917-6
Date
2018
Publisher
Springer
Degree
Supervisors
Rights
© 2018 Springer US. This is the author's accepted version. The final publication is available at Springer via dx.doi.org/10.1007/s00233-018-9917-6.