Groupoids of left quotients

TL;DR

This paper characterizes left orders and q-orders within groupoids, providing insights into their structure and connections to inverse semigroups, advancing the theoretical understanding of algebraic orderings.

Contribution

It offers a new characterization of left orders and q-orders in groupoids and explores their relationship with left I-orders in primitive inverse semigroups.

Findings

Characterization of left orders in groupoids

Description of q-orders in groupoids

Relationship between left I-orders and groupoid orders

Abstract

A subcategory $\textbf{C}$ 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 $a, b \in \textbf{C}$. 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.