Algebras with Medial-like Functional Equations on Quasigroups

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 $(B; +)$.

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