Algebras with Medial-like Functional Equations on Quasigroups

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 \((B; +)\).