Groupoids of left quotients

Abstract

A subcategory $\text{C}$ of a groupoid $\mathbb{G}$ is a left order in $\mathbb{G}$, if every element of $\mathbb{G}$ can be written as $a^{-1}b$ where $a,b\in \text{C}$. A subsemigroupoid $\mathfrak{C}$ of a groupoid $\mathbb{G}$ is a left q-order in $\mathbb{G}$, if every element of $\mathbb{G}$ can be written as $a^{-1}b$ where $a,b\in \mathfrak{C}$. We give a characterization of left orders (q-orders) in groupoids. In addition, we describe the relationship between left I-orders in primitive inverse semigroups and left orders (q-orders) in groupoids.