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A Characterization of Identities Implying Congruence Modularity I

Published online by Cambridge University Press:  20 November 2018

Alan Day  and
Ralph Freese
Alan Day
Affiliation:
Lakehead University, Thunder Bay, Ontario
Ralph Freese
Affiliation:
University of Hawaii, Honolulu, Hawaii
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In his thesis and [24], J. B. Nation showed the existence of certain lattice identities, strictly weaker than the modular law, such that if all the congruence lattices of a variety of algebras satisfy one of these identities, then all the congruence lattices were even modular. Moreover Freese and Jónsson showed in [10] that from this “congruence modularity” of a variety of algebras one can even deduce the (stronger) Arguesian identity.

These and similar results [3; 5; 9; 12; 18; 21] induced Jónsson in [17; 18] to introduce the following notions. For a variety of algebras , is the (congruence) variety of lattices generated by the class () of all congruence lattices θ(A), . Secondly if is a lattice identity, and Σ is a set of such, holds if for any variety implies .

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 5 , 01 October 1980 , pp. 1140 - 1167
Copyright
Copyright © Canadian Mathematical Society 1980

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