Bicyclic semigroups of left I-quotients

TL;DR

This paper investigates the structure of left I-orders within the bicyclic monoid, providing conditions for their characterization and proving their straightness property.

Contribution

It establishes necessary and sufficient conditions for subsemigroups to be left I-orders in the bicyclic monoid and proves all such orders are straight.

Findings

Characterization of left I-orders in the bicyclic monoid

Necessary and sufficient conditions for subsemigroups to be left I-orders

Proof that all left I-orders in the bicyclic monoid are straight

Abstract

In this article we study left I-orders in the bicyclic monoid $\mathcal{B}$. We give necessary and sufficient conditions for a subsemigroup of $\mathcal{B}$ to be a left I-oreder in $\mathcal{B}$. We then prove that any left I-order in $\mathcal{B}$ is straight.