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We study the endomorphisms of idempotent medial quasigroups and determinability of some classes of medial quasigroups by their endomorphisms. We introduce the endomorphism algebra of idempotent medial quasigroup and prove that if the endomorphism algebras of quasigroups are isomorphic, then the corresponding quasigroups are isomorphic as well. In addition, we present a counterexample to demonstrate that the endomorphism algebras can not be replaced by the endomorphism monoids.

On determinability of some classes of medial quasigroups by their endomorphisms