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\title{\bf Categorical Equivalence of Varieties and Invariant Relations} 
\author{K. Denecke and O. Lders }
%\date{~}
\maketitle
\begin{center}
{\bf   ABSTRACT}
\end{center}
Each variety of algebras of a given type can be considered as a 
category, where the objects are the algebras in the variety, while the 
morphisms are the homomorphisms between algebras. In applications the
weaker concept of an {\it equivalence} between categories is used more often 
than the concept of an isomorphism. For instance, the category
${\cal C}$ of finite-dimensional vector spaces over the field $K$ is 
equivalent with the category of all vector spaces which are dual to the 
vector spaces from ${\cal C}$.

For two varieties $V$ and $W$ to be {\it categorically equivalent} 
means precisely the following: there exists a covariant functor $F$ from $V$ 
to $W$ such that \\[-8mm]
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item for all ${\underline A}, {\underline B} \in V$ the functor $F$ defines 
      a bijection between $hom({\underline A}, {\underline B})$
\item for all ${\underline C} \in W$ there is an algebra ${\underline A} \in
      V$ such that $F({\underline A})$ is isomorphic to ${\underline C}$.
\end{enumerate}

We may remark that if $V$ and $W$ are categorically equivalent under 
the categorical equivalence $F$
then ${\underline A}$ is isomorphic to ${\underline B}$ iff $F({\underline 
A})$ is isomorphic to $F({\underline B})$ whenever ${\underline A}, 
{\underline B} \in V$. 

For varieties of $R$-modules the problem of an algebraic characterization of 
categorical equivalences was solved a long time ago:
The variety of $R$-modules is categorically equivalent with the variety of 
$S$-modules iff the rings $R$ and $S$ are Morita equivalent.
 In Universal Algebra the best known and most important example of a 
categorical equivalence is that one between the variety of Boolean algebras 
(generated by the two-element Boolean algebra) and any variety generated by 
a single primal algebra, i.e. a finite algebra where arbitrary operations on 
the universe are term operations of the algebra ([Hu; 69]). This example 
shows that from an algebraic point of view categorically equivalent 
varieties can have quite different properties.

In this paper we will give a different characterization of categorical 
equivalent varieties using invariant relations. 

When the authors started in the late eighties to generalize the categorical 
equivalence of T. K. Hu between the variety of Boolean algebras and the 
varieties generated by an arbitrary finite algebra to varieties generated by
arbitrary two-element algebras ([Den; 85]) they used 
duality theory and the ``Equivalent Prevarieties Theorem'' of B. A. Davey 
and H. Werner ([Dav-W; 80]). In 1992 R. McKenzie ([McK; 96]) gave a complete algebraic
characterization of categorical equivalent varieties using the matrix power construction and
idempotent invertible terms. In [Den-L; 95] the authors started to use the Galois-connection
between term operations of an algebra and its invariant relations. Our main result is that
two varieties which are generated by finite algebras are categorically equivalent iff their
algebras of invariant relations are isomorphic. Moreover, we will give several applications.
\vspace{1cm}

\section*{References}
\begin{description}
\item[[Dav-W; 80]] B. A. Davey, H.Werner, Dualities and equivalences for 
         varieties of algebras, in: Contributions to Lattice Theory (Proc. 
         Conf. Szeged 1980), Colloq. Math. Soc. J\'anos Bolyai, No 33, 
         North-Holland, pp. 101-275
\item[[Den-L; 95]] K. Denecke, O. Lders, Category equivalences of clones,
         Algebra Universalis \underline{34} (1995), 608-618
\item[[McK; 96]] R. McKenzie, An algebraic version of categorical 
         equivalence for vayrieties and more general algebraic theories, in: 
         A. Ursini and P. Agliano (editors): Logic aynd Algebra, Vol. 180 of 
         Lecture Notes in Pure and Applied Mathematics, pp. 211-243, Marcel 
         Dekker, 1996
\end{description}



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