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Congruence intersection properties for varieties of algebras

Part of: Varieties Algebraic structures

Published online by Cambridge University Press:  09 April 2009

Paolo Agliano  and
Kirby A. Baker
Paolo Agliano
Affiliation:
Dipartimento di Matematica Via del Capitano 15 53100 Siena Italy e-mail: agliano@unisi.it
Kirby A. Baker
Affiliation:
Department of Mathematics UCLA Los Angeles CA 90095-1555 USA e-mail: baker@math.ucla.edu
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Abstract

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It is shown that a variety ν has distributive congruence lattices if and only if the intersection of two principal congruence relations is definable by equations involving terms with parameters. The nature of the terms involved then provides a useful classification of congruence distributive varieties. In particular, the classification puts into proper perspective two stronger properties. A variety is said to have the Principal Intersection Property if the intersection of any two principal congruence relations is principal, or the Compact Intersection Property if the intersection of two compact congruence relations is compact. For non-congruence-distributive varieties, it is shown that some useful constuctions are nevertheless possible.

Keywords

principal intersection propertycompact intersection property.

MSC classification

Secondary: 08A30: Subalgebras, congruence relations 08B10: Congruence modularity, congruence distributivity
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 67 , Issue 1 , August 1999 , pp. 104 - 121
Copyright
Copyright © Australian Mathematical Society 1999

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