Let D be a finite distributive lattice with n join-irreducible elements. It is well-known that D can be represented as the congruence lattice of a rectangular lattice L which is a special planer semimodular lattice. In this paper, we shall give a better upper bound for the size of L by a function of n, improving a 2009 result of G. Grätzer and E. Knapp.
GrätzerG. and KnappE., Notes on planar semimodular lattices. II. Congruences, Acta Sci Math (Szeged)74 (2008), 37–47.
5.
GrätzerG. and KnappE., Notes on planar semimodular lattices. III. Rectangular lattices, Acta Sci Math (Szeged)75 (2009), 29–48.
6.
GrätzerG. and KnappE., Notes on planar semimodular lattices. IV. The size of a minimal congruence lattices representation with rectangular lattices, Acta Sci Math (Szeged)76 (2010), 3–26.
7.
GrätzerG. and LakserH., Homomorphisms of distributive lattices as restrictions of congrunces, Canad J Math38 (1986), 1122–1134.
8.
GrätzerG. and LakserH., Congruence lattices of planar lattices, Acta Math Sci Hungar60 (1992), 251–268.
9.
GrätzerG., LakserH. and SchmidtE.T., Congruence lattices of small planar lattices, Proc Amer Math Soc123 (1995), 2619–2623.
10.
GrätzerG., LakserH. and SchmidtE.T., Congruence lattices of finite semimodular lattices, Canad Math Bull41 (1998), 290–297.
11.
GrätzerG., RivalI. and ZaguiaN., Small representations of finite distributive lattices as congruence lattices, Proc Amer Math Soc123 (1998), 1959–1961. Correction: 126 (1998) 2509-2510.
12.
GrätzerG. and SchmidtE.T., On congruence lattices of lattices, Acta Math Sci Hungar13 (1962), 179–185.
13.
GrätzerG. and WehrungF., Lattice theory: special topics and applications, Birkhäuser, Basel, 2014.
