Some properties of Neumann quasigroups

TL;DR

This paper explores properties of Neumann quasigroups, showing they can be represented via abelian groups, and establishes their equivalence with Schweizer quasigroups, including automorphism group characteristics.

Contribution

It characterizes Neumann quasigroups in terms of abelian groups and proves their equivalence with Schweizer quasigroups, expanding understanding of their algebraic structure.

Findings

Neumann quasigroups can be represented as x - y over abelian groups.

Automorphism group of Neumann quasigroups equals automorphisms of the underlying abelian group.

Neumann quasigroups are equivalent to Schweizer quasigroups and are GA-quasigroups.

Abstract

Any Neumann quasigroup $(Q, \cdot)$ (quasigroup with Neumann identity $ x \cdot(yz \cdot yx) = z$ is called Neumann quasigroup) can be presented in the form $x\cdot y = x-y$, where $(Q, +)$ is an abelian group. Automorphism group of Neumann quasigroup coincides with the group $Aut(Q, +)$. Any Schweizer quasigroup (quasigroup with Schweizer identity $xy \cdot xz = zy$ is called Schweizer quasigroup) is a Neumann quasigroup and vice versa. Any Neumann quasigroup is a GA-quasigroup.