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Tolerances and Commutators on lattices

Published online by Cambridge University Press:  17 April 2009

Dietmar Schweigert
Dietmar Schweigert
Affiliation:
FB Mathematik, der Universität, D-6750 Kaiserslautern, Fed. Rep., Germany
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Abstract

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The commutator has the following order theoretic properties: [α, β] ≦ α ∧ β, [α, β] = [β α],[α1 ∨ α2,β] = [α1, β] ∨ [α2, β] for congruences α, β ∈ Con A of an algebra A in a congruence modular variety generalising the original concept in group theory. A tolerance of a lattice L is a reflexive and symmetric sublattice of L2. We show that to every commutator [ , ] of Con A corresponds a ∧-subsemilattice of the lattice of tolerances of Con A. It can be shown that A in a congruence modular variety is nilpotent if |con A| > 2 and Con A is simple.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 37 , Issue 2 , April 1988 , pp. 213 - 219
Copyright
Copyright © Australian Mathematical Society 1988

References

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