Semigroup Forum

, Volume 96, Issue 2, pp 377–395 | Cite as

Left and right negatively orderable semigroups and a one-sided version of Simon’s theorem

  • Zsófia JuhászEmail author
Research Article


We study the classes \(\mathrm {LNO}\) and \(\mathrm {RNO}\) of left and right negatively orderable semigroups, arising as natural one-sided generalizations of negatively orderable semigroups (\(\mathrm {NO}\)). Negatively ordered monoids are well-known, for instance, from an equivalent formulation by Straubing and Thérien, of a celebrated theorem by I. Simon on piecewise testable languages. The main aim of this paper is to prove a one-sided version of Simon’s theorem for \(\mathrm {LNO}\) (\(\mathrm {RNO}\)). Analogues of some well-known results about negatively orderable semigroups are established in these cases. A characterisation of right negatively orderable semigroups as semigroups of certain type of decreasing mappings on partially ordered sets is obtained. We present sets of quasi-identities defining the quasivarieties \(\mathrm {LNO}\) and \(\mathrm {RNO}\) and show that these classes cannot be defined by finite sets of quasi-identities. We prove \(\mathrm {NO}=\mathrm {LNO}\cap \mathrm {RNO}\) and describe \(\mathrm {LNO}\vee \mathrm {RNO}\). As a one-sided version of Simon’s theorem, the pseudovariety generated by all finite semigroups in \(\mathrm {LNO}\) (\(\mathrm {RNO}\)) is determined and an Eilenberg-type correspondence between this pseudovariety and a variety of languages is established.


Semigroup Language Pseudovariety Ordered semigroup Quasiorder Syntactic semigroup 


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Authors and Affiliations

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

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