Categorical equivalence of varieties and invariant relations. (Q1771863)

Varieties of algebras are considered as categories where morphisms are homomorpisms. Two varieties \({\mathcal V}, {\mathcal W}\) are categorically equivalent if there exists a covariant functor \(F\) from \(\mathcal V\) to \(\mathcal W\) such that for all \(A,B\in {\mathcal V}\) the functor \(F\) defines a bijection between \(\Hom (A,B)\) and \(\Hom(F(A),F(B))\) and for each \(C\in {\mathcal V}\) there is \(A\in {\mathcal V}\) such that \(F(A)\) is isomorphic to \(C\). It is shown that two varieties \({\mathcal V} (A),\, {\mathcal V} (B)\) generated by finite algebras \(A\) and \(B\), respectively, are categorically equivalent if and only if the algebras of invariant relations of the clones of all term operations of \(A\) and \(B\) are isomorphic.