Categorical equivalence of varieties and invariant relations

Abstract.

Two varieties $V({\underline A}), V({\underline B})$ generated by finite algebras ${\underline A}$ and $\underline B$, respectively are categorically equivalent under an equivalence functor which takes $\underline A$ to $\underline B$ iff the algebras of invariant relations of the clones of all term operations of $\underline A$ and $\underline B$ are isomorphic. In this paper we will prove this theorem and will give several applications.