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Congruences and prime-perspectivities in finite lattices

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Abstract

In a finite lattice, a congruence spreads from a prime interval to another by a sequence of congruence-perspectivities through intervals of arbitrary size, by a 1955 result of J. Jakubík.

In this note, I introduce the concept of prime-perspectivity and prove the Prime-projectivity Lemma: a congruence spreads from a prime interval to another by a sequence of prime-perspectivities through prime intervals.

A planar semimodular lattice is slim, if it contains no M 3 sublattice. I introduce the Swing Lemma, a very strong version of the Prime-projectivity Lemma for slim, planar, and semimodular lattices.

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Authors and Affiliations

  1. Department of Mathematics, University of Manitoba, Winnipeg, MB, R3T 2N2, Canada

    G. Grätzer

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  1. G. Grätzer
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Correspondence to G. Grätzer.

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Presented by R. Quackenbush.

Dedicated to Brian Davey on the occasion of his 65th birthday

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Grätzer, G. Congruences and prime-perspectivities in finite lattices. Algebra Univers. 74, 351–359 (2015). https://doi.org/10.1007/s00012-015-0355-y

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  • DOI: https://doi.org/10.1007/s00012-015-0355-y

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