Semigroups of straight left inverse quotients

Abstract

Let $Q$ be an inverse semigroup. A subsemigroup $S$ of $Q$ is a left I-order in $Q$ and $Q$ is a semigroup of left I-quotients of $S$ 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 being able to take $a$ and $b$ to be $\mathcal{R}$-related in $Q$ we say that $S$ is straight in $Q$ and $Q$ is a semigroup of straight left I-quotients of $S$. We give a set of necessary and sufficient conditions for a semigroup to be a straight left I-order. The conditions are in terms of two binary relations, corresponding to the potential restrictions of $\mathcal{R}$ and $\mathcal{L}$ from an oversemigroup, and an associated partial order. Our approach relies on the meet structure of the $\mathcal{L}$ of inverse semigroups. We prove that every finite left I-order is straight and give an example of a left I-order which is not straight.