Congruences and prime-perspectivities in finite lattices

Abstract

IIn a finite lattice, a congruence spreads from a prime interval to another by a sequence of congruence-perspectivities through \emph{intervals of arbitrary size}, by a 1955 result of J. Jakub\'ik. In this note, I introduce the concept of \emph{prime-perspectivity} and prove the Prime-projectivity Lemma: a congruence spreads from a prime interval to another by a sequence of prime-perspectivities through \emph{prime ntervals}. A planar semimodular lattice is \emph{slim} if it contains no $\mathsf{M}_3$ sublattice. I introduce the Swing Lemma, a very strong version of the Prime-projectivity Lemma for slim, planar, semimodular lattices.