Congruences on bicyclic extensions of a linearly ordered group

Abstract

In the paper we study inverse semigroups $\mathcal{B}(G)$, ${\mathcal{B}}^{+}(G)$, $\bar{\mathcal{B}}(G)$ and $\bar{\mathcal{B}}{}^{+}(G)$ 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 $\mathcal{B}(G)$, ${\mathcal{B}}^{+}(G)$, $\bar{\mathcal{B}}(G)$ and $\bar{\mathcal{B}}{}^{+}(G)$, their bands and show that they are simple, and moreover the semigroups $\mathcal{B}(G)$ and ${\mathcal{B}}^{+}(G)$ are bisimple. We show that for a commutative linearly ordered group $G$ all non-trivial congruences on the semigroup $\mathcal{B}(G)$ (and ${\mathcal{B}}^{+}(G)$) are group congruences if and only if the group $G$ is archimedean. Also we describe the structure of group congruences on the semigroups $\mathcal{B}(G)$, ${\mathcal{B}}^{+}(G)$, $\bar{\mathcal{B}}(G)$ and $\bar{\mathcal{B}}{}^{+}(G)$.

20 pages