## Journal of Symbolic Logic

- J. Symbolic Logic
- Volume 59, Issue 3 (1994), 912-923.

### Connections between Axioms of Set Theory and Basic Theorems of Universal Algebra

H. Andreka, A. Kurucz, and I. Nemeti

#### Abstract

One of the basic theorems in universal algebra is Birkhoff's variety theorem: the smallest equationally axiomatizable class containing a class $\mathbf{K}$ of algebras coincides with the class obtained by taking homomorphic images of subalgebras of direct products of elements of $\mathbf{K}$. G. Gratzer 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 $ZF + AC \{\text{Foundation}\}$. 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 $ZF \{\operatorname{Foundation}$. The second part of the paper deals with further connections between axioms of $ZF$-set theory and theorems of universal algebra.

#### Article information

**Source**

J. Symbolic Logic, Volume 59, Issue 3 (1994), 912-923.

**Dates**

First available in Project Euclid: 6 July 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.jsl/1183744557

**Mathematical Reviews number (MathSciNet)**

MR1295978

**Zentralblatt MATH identifier**

0808.03036

**JSTOR**

links.jstor.org

#### Citation

Andreka, H.; Kurucz, A.; Nemeti, I. Connections between Axioms of Set Theory and Basic Theorems of Universal Algebra. J. Symbolic Logic 59 (1994), no. 3, 912--923. https://projecteuclid.org/euclid.jsl/1183744557