A note on semisymmetry

TL;DR

This paper surveys properties of a semisymmetrization functor in quasigroup categories, introduces a new functor mapping quasigroups to their squares, and compares it to previous approaches.

Contribution

It proposes a new semisymmetrization functor that maps quasigroups to their squares, offering an alternative to the previous cube-mapping functor.

Findings

The survey clarifies properties of the existing semisymmetrization functor.

A new functor mapping quasigroups to their squares is introduced.

Comparison shows the new functor differs from the previous cube-mapping approach.

Abstract

A survey of properties of the adjunction involving a semisymmetrization functor, which was suggested by J.D.H. Smith, and which maps the category of quasigroups with homotopies to the category of semisymmetric quasigroups with homomorphisms, is given. A new semisymmetrization functor is suggested. This functor maps a quasigroup to its square instead to its cube as it was the case with the former functor.