Bisimple inverse $\omega$-semigroups of left I-quotients

TL;DR

This paper characterizes when a semigroup can serve as a left I-order within bisimple inverse ω-semigroups, focusing on the algebraic structure and conditions for such embeddings.

Contribution

It provides necessary and sufficient conditions for a semigroup to be a left I-order in bisimple inverse ω-semigroups, advancing the understanding of their algebraic structure.

Findings

Established criteria for semigroups to be left I-orders in bisimple inverse ω-semigroups.

Connected the concept of straight left I-orders with algebraic properties of inverse semigroups.

Enhanced the classification of inverse semigroups via I-order structures.

Abstract

A subsemigroup $S$ of an inverse semigroup $Q$ is a left I-order in $Q$ if every element in $Q$ can be written as $a^{-1}b$ where $a,b \in S$ and $a^{-1}$ is the inverse of $a$ in the sense of inverse semigroup theory. If we insist on $a$ and $b$ being $\mathcal{R}$-related in $Q$, then we say that $S$ is a straight left I-order in $Q$. We give necessary and sufficient conditions for a semigroup to be a left I-order in a bisimple inverse $\omega$-semigroup.