AG-groups and other classes of right Bol quasigroups

TL;DR

This paper explores the relationships between AG-groups, Bol* quasigroups, and right Bol quasigroups, showing how they can be derived from abelian groups and groups respectively, and provides enumeration results for these classes.

Contribution

It introduces Bol* quasigroups and demonstrates their connection to groups, extending Sharma's correspondence to new algebraic structures.

Findings

AG-groups are obtained from abelian groups via involutive automorphisms.

Bol* quasigroups correspond to groups, similar to right Bol quasigroups.

Enumeration results for AG-groups, Bol* quasigroups, and right Bol quasigroups are provided.

Abstract

By a result of Sharma, right Bol quasigroups are obtainable from right Bol loops via an involutive automorphism. We prove that the class of AG-groups, introduced by Kamran, is obtained via the same construction from abelian groups. We further introduce a new class of Bol* quasigroups, which turns out to correspond, as above, to the class of groups. Sharma's correspondence allows an efficient implementation and we present some enumeration results for the above three classes.