CONNECTIONS BETWEEN AXIOMS OF SET-THEORY AND BASIC THEOREMS OF UNIVERSAL ALGEBRA

One of the basic theorems in universal algebra is Birkhoffs variety th eorem: the smallest equationally axiomatizable class containing a clas s K of algebras coincides with the class obtained by taking homomorphi c images of subalgebras of direct products of elements of K. G. Gratze r asked whether the variety theorem is equivalent to the Axiom of Choi ce. In 1980. two of the present authors proved that Birkhoffs theorem can already be derived in ZF. Surprisingly. the Axiom of Foundation pl ays a crucial role here: we show that Birkhoffs theorem cannot be deri ved in ZF + AC\{Foundation}. even if we add Foundation for Finite Sets . We also prove that the variety theorem is equivalent to a purely set -theoretical statement, the Collection Principle. This principle is in dependent of ZF\{Foundation}. The second part of the paper deals with further connections between axioms of ZF-set theory and theorems of un iversal algebra.