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Connections between axioms of set theory and basic theorems of universal algebra

Published online by Cambridge University Press:  12 March 2014

H. Andréka ,
Á. Kurucz  and
I. Németi
H. Andréka
Affiliation:
Mathematical Institute of the Hungarian Academy of Sciences, Budapest, H-1364, Hungary, E-mail: [email protected]
Á. Kurucz
Affiliation:
Mathematical Institute of the Hungarian Academy of Sciences, Budapest, H-1364, Hungary, E-mail: [email protected]
I. Németi
Affiliation:
Mathematical Institute of the Hungarian Academy of Sciences, Budapest, H-1364, Hungary, E-mail: [email protected]
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Abstract.

One of the basic theorems in universal algebra is Birkhoff's variety theorem: the smallest equationally axiomatizable class containing a class K of algebras coincides with the class obtained by taking homomorphic images of subalgebras of direct products of elements of K. G. Grätzer asked whether the variety theorem is equivalent to the Axiom of Choice. In 1980, two of the present authors proved that Birkhoff's theorem can already be derived in ZF. Surprisingly, the Axiom of Foundation plays a crucial role here: we show that Birkhoff's theorem cannot be derived 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 independent of ZF\{Foundation}. The second part of the paper deals with further connections between axioms of ZF-set theory and theorems of universal algebra.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 59 , Issue 3 , September 1994 , pp. 912 - 923
Copyright
Copyright © Association for Symbolic Logic 1994

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References

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