Dimension and partial groups

TL;DR

This paper explores the dimensional properties of partial groups and groupoids, establishing that finite partial groups are genuinely finite and characterizing their dimensions in relation to groups.

Contribution

It introduces the concept of dimension for partial groups and groupoids, showing finite partial groups are genuinely finite and linking dimension to group structure.

Findings

Finite partial groups are genuinely finite despite infinite data representation.

Dimension of a partial group equals its size if and only if it is a group.

Finite partial groups have finitely many subgroups.

Abstract

A partial group with $n+1$ elements is, when regarded as a symmetric simplicial set, of dimension at most $n$. This dimension is $n$ if and only if the partial group is a group. As a consequence of the first statement, finite partial groups are genuinely finite, despite being seemingly specified by infinitely much data. In particular, finite partial groups have only finitely many im-partial subgroups. We also consider dimension of partial groupoids.