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Left and right negatively orderable semigroups where every element has a left and a right identity

  • Zsófia Juhász
Research Article
  • 51 Downloads

Abstract

We study left and right negatively orderable semigroups, natural one-sided generalizations of negatively orderable semigroups, in the wide class \({\mathfrak {C}}\) of semigroups where every element has a left and a right identity. In semigroups in \({\mathfrak {C}}\) the smallest left and smallest right negatively orderable congruences \(\theta _{{ }_{{\mathrm {LNO}}}}\) and \(\theta _{{ }_{{\mathrm {RNO}}}}\) commute; furthermore, we have: \(\theta _{{ }_{{\mathrm {LNO}}}}\circ \theta _{{ }_{{\mathrm {RNO}}}}=\theta _{{ }_{{\mathrm {RNO}}}}\circ \theta _{{ }_{{\mathrm {LNO}}}}=\theta _{{ }_{{\mathrm {RNO}}}}\vee \theta _{{ }_{{\mathrm {LNO}}}}=\theta _{{ }_{{\mathbf {{\mathrm {NO}}}}}}\), where \(\theta _{{ }_{{\mathbf {{\mathrm {NO}}}}}}\) is the smallest negatively orderable congruence. These properties give rise to the following decomposition: for every semigroup S in \({\mathfrak {C}}\), \(S/(\theta _{{ }_{{\mathrm {LNO}}}}\cap \theta _{{ }_{{\mathrm {RNO}}}})\) is isomorphic to the spined product of the largest left and the largest right negatively orderable images \(S/\theta _{{ }_{{\mathrm {LNO}}}}\) and \(S/\theta _{{ }_{{\mathrm {RNO}}}}\) of S over its largest negatively orderable image \(S/\theta _{{ }_{{\mathbf {{\mathrm {NO}}}}}}\). Descriptions of finite left (right) negatively orderable semigroups and of \((\theta _{{}_{{\mathrm {LNO}}}}\cap \theta _{{}_{{\mathrm {RNO}}}})\)-trivial semigroups in \({\mathfrak {C}}\) are obtained and the pseudovarieties generated by these semigroups are determined. A regular semigroup is left (right) negatively orderable if and only if it is a right (left) normal band. Counterexamples of semigroups where \(\theta _{{ }_{{\mathrm {LNO}}}}\circ \theta _{{ }_{{\mathrm {RNO}}}} \ne \theta _{{ }_{{\mathrm {RNO}}}}\circ \theta _{{ }_{{\mathrm {LNO}}}}\) and \(\theta _{{ }_{{\mathrm {LNO}}}}\vee \theta _{{ }_{{\mathrm {RNO}}}} \ne \theta _{{ }_{{\mathbf {{\mathrm {NO}}}}}}\) are presented.

Keywords

Semigroup Quasiorder Spined product Pseudovariety Ordered semigroup 

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Authors and Affiliations

  1. 1.Department of Computer Algebra, Faculty of InformaticsEötvös Loránd UniversityBudapestHungary

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