Congruences on bicyclic extensions of a linearly ordered group

TL;DR

This paper investigates the algebraic structure of certain inverse semigroups generated by partial monotone translations of a linearly ordered group, providing descriptions of Green's relations, simplicity, bisimplicity, and congruences.

Contribution

It characterizes Green's relations, simplicity, and congruences of semigroups generated by partial translations of linearly ordered groups, including the structure of group congruences.

Findings

Semigroups are simple and bisimple under certain conditions.

Green's relations are explicitly described for these semigroups.

Non-trivial congruences are group congruences if and only if the group is archimedean.

Abstract

In the paper we study inverse semigroups $\mathscr{B}(G)$, $\mathscr{B}^+(G)$, $\bar{\mathscr{B}}(G)$ and $\bar{\mathscr{B}}\,^+(G)$ which are generated by partial monotone injective translations of a positive cone of a linearly ordered group $G$. We describe Green's relations on the semigroups $\mathscr{B}(G)$, $\mathscr{B}^+(G)$, $\bar{\mathscr{B}}(G)$ and $\bar{\mathscr{B}}\,^+(G)$, their bands and show that they are simple, and moreover the semigroups $\mathscr{B}(G)$ and $\mathscr{B}^+(G)$ are bisimple. We show that for a commutative linearly ordered group $G$ all non-trivial congruences on the semigroup $\mathscr{B}(G)$ (and $\mathscr{B}^+(G)$) are group congruences if and only if the group $G$ is archimedean. Also we describe the structure of group congruences on the semigroups $\mathscr{B}(G)$, $\mathscr{B}^+(G)$, $\bar{\mathscr{B}}(G)$ and $\bar{\mathscr{B}}\,^+(G)$.