AG-groups as parallelogram spaces

TL;DR

This paper proves that the parallelogram space of an AG-group is itself an AG-group, extending the result to medial quasigroups and providing methods to find vertices of parallelograms.

Contribution

It offers a direct proof that the parallelogram space of an AG-group is an AG-group and generalizes this to medial quasigroups, with practical vertex-finding methods.

Findings

Parallelogram space of an AG-group is an AG-group.

For Abelian groups, the parallelogram space remains Abelian.

Provides methods to find other vertices of a parallelogram.

Abstract

It is known that an AG-group is paramedial and a paramedial is a parallelogram space. From which it follows that an AG-group is a parallelogram space. In this paper we give a direct proof of this fact and study it further. Our main result is that the parallelogram space of an AG-group is again an AG-group, which particularly implies that the parallelogram space for an Abelian group is also an Abelian group. We then generalise this result to medial quasigroups. Finally, we provide some quick methods of finding the other vertices of this parallelogram if at least one nontrivial vertex is known.