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A model theoretic approach to Malcev conditions

Published online by Cambridge University Press:  12 March 2014

John T. Baldwin  and
Joel Berman
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A variety V (equational class of algebras) satisfies a strong Malcev condition ∃f1,…, ∃fnθ(f1, …, fn, x1, …, xm) where θ is a conjunction of equations in the function variables f1, …, fn and the individual variables x1, …, xm, if there are polynomial symbols p1, …, pn in the language of V such that ∀x1, …, xmθ(p1 …, pn, x1, …, xm) is a law of V. Thus a strong Malcev condition involves restricted second order quantification of a strange sort. The quantification is restricted to functions which are “polynomially definable”. This notion was introduced by Malcev [6] who used it to describe those varieties all of whose members have permutable congruence relations. The general formal definition of Malcev conditions is due to Grätzer [1]. Since then and especially since Jónsson's [3] characterization of varieties with distributive congruences there has been extensive study of strong Malcev conditions and the related concepts: Malcev conditions and weak Malcev conditions.

In [9], Taylor gives necessary and sufficient semantic conditions for a class of varieties to be defined by a (strong) Malcev condition. A key to the proof is the translation of the restricted second order concepts into first order concepts in a certain many sorted language. In this paper we show that, given this translation, Taylor's theorem is an easy consequence of a result of Tarski [8] and the standard preservation theorems of first order logic.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 42 , Issue 2 , June 1977 , pp. 277 - 288
Copyright
Copyright © Association for Symbolic Logic 1977

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References

REFERENCES

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