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Local Malcev Conditions

Published online by Cambridge University Press:  20 November 2018

Alden F. Pixley
Alden F. Pixley*
Affiliation:
Harvey Mudd College, the Claremont Colleges, Claremont, California
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Abstract

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Let p and q be polynomial symbols of a type of algebras having operations ∨, ∧, and; (interpreted as the join, meet, and product of congruence relations). If is an algebra, L(), the local variety of , is the class of all algebras such that for each finite subset G of there is a finite subset F of such that every identity of F is also an identity of G.

THEOREM. There is an algorithm which, for each inequality

p≤q,

and pair of integers n, k≥2, determines a set Un, k of (Malcev) equations with the property:

For each algebra, p≤q is true in the congruence lattice offor each∊L() if and only if for each finite subset F ofand integer n≥2 there is a k=k(n, F) such that Un, kare identities of F.

This generalizes a corresponding result for varieties due to Wille (Kongruenzklassengeometrien, Lect. Notes in Math. Springer- Verlag, Berlin-Heidelberg, New York, 1970) and at the same time provides a more direct proof.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 15 , Issue 4 , December 1972 , pp. 559 - 568
Copyright
Copyright © Canadian Mathematical Society 1972

References

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