Algebras with Medial-like Functional Equations on Quasigroups

TL;DR

This paper investigates 14 medial-like functional equations in quasigroup algebras and proves that such algebras can be represented linearly over an abelian group, revealing structural properties of these algebraic systems.

Contribution

It introduces a classification of medial-like equations in quasigroups and establishes a linear representation theorem for algebras satisfying these equations.

Findings

Algebras with medial-like equations have linear representations on abelian groups.

14 specific medial-like balanced functional equations are considered.

Structural characterization of quasigroup algebras satisfying these equations.

Abstract

We consider $14$ medial-like balanced functional equations with four object variables for a pair $(f, g)$ of binary quasigroup operations. Then, we prove that every algebra $(B; f, g)$ with quasigroup operations satisfying a medial-like balanced functional equation has a linear representation on an abelian group $(B; +)$.