ACTA issues

Lamps in slim rectangular planar semimodular lattices

Gábor Czédli

Acta Sci. Math. (Szeged) 87:3-4(2021), 381-413

Abstract. A planar (upper) semimodular lattice $L$ is \emph {slim} if the five-element nondistributive modular lattice $M_3$ does not occur among its sublattices. (Planar lattices are finite by definition.) \emph {Slim rectangular lattices} as particular slim planar semimodular lattices were defined by G. Grätzer and E. Knapp in 2007. In 2009, they also proved that the congruence lattices of slim planar semimodular lattices with at least three elements are the same as those of slim rectangular lattices. In order to provide an effective tool for studying these congruence lattices, we introduce the concept of \emph {lamps} of slim rectangular lattices and prove several of their properties. Lamps and several tools based on them allow us to prove in a new and easy way that the congruence lattices of slim planar semimodular lattices satisfy the two previously known properties. Also, we use lamps to prove that these congruence lattices satisfy four new properties including the \emph {Two-pendant Four-crown Property} and the \emph {Forbidden Marriage Property}.

DOI: 10.14232/actasm-021-865-y

AMS Subject Classification (1991): 06C10

Keyword(s): rectangular lattice, slim semimodular lattice, multifork extension, lattice diagram, edge of normal slope, precipitous edge, lattice congruence, two-pendant four-crown property, lamp, congruence lattice, forbidden marriage property

received 15.1.2021, revised 5.3.2021, accepted 11.3.2021. (Registered under 115/2021.)