Embeddable partial groups

TL;DR

This paper explores conditions under which partial groups can be embedded into groups, providing a folklore theorem, examining non-embeddable examples, and extending results to partial groupoids and groupoids.

Contribution

It formalizes a folklore theorem on partial group embeddability and characterizes non-embeddable partial groups and groupoids, extending the theory.

Findings

A partial group embeds in a group iff each word has at most one multiplication.

Characterization of non-embeddable partial groups.

A partial groupoid embeds in a groupoid iff its reduction embeds in a group.

Abstract

We record a folklore theorem that says a partial group embeds in a group if and only if each word has at most one possible multiplication, regardless of choice of parenthesization. We further investigate the partial groups which are exemplars of non-embeddability. Finally we show that a partial groupoid embeds in a groupoid if and only if its reduction embeds in a group.