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

Abstract

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.