Congruences and prime-perspectivities in finite lattices

2013/12/09 by G. Grätzer, Grätzer, G.
Mathematics · #06B10 #FOS: Mathematics #Rings and Algebras (math.RA) #math.RA #msc:06B10

paper · pdf · doi:10.48550/arxiv.1312.2537

arxiv created 2014/11/17 · arxiv updated 2014/11/18

Abstract

IIn 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 ntervals. A planar semimodular lattice is slim if it contains no M₃ sublattice. I introduce the Swing Lemma, a very strong version of the Prime-projectivity Lemma for slim, planar, semimodular lattices.

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