CD-independent subsets in meet-distributive lattices

Abstract

A subset X of a finite lattice L is CD-independent if the meet of any two incomparable elements of X equals 0. In 2009, Czédli, Hartmann and Schmidt proved that any two maximal CD-independent subsets of a finite distributive lattice have the same number of elements. In this paper, we prove that if L is a finite meet-distributive lattice, then the size of every CDindependent subset of L is at most the number of atoms of L plus the length of L. If, in addition, there is no three-element antichain of meet-irreducible elements, then we give a recursive description of maximal CD-independent subsets. Finally, to give an application of CD-independent subsets, we give a new approach to count islands on a rectangular board.Date: July 9, 2013 2000 Mathematics Subject Classification. Primary 06C10; secondary 05A05 and 05B25 .

Cardinality of height function’s range in case of maximally many rectangular islands — computed by cuts

“…The problem of minimum cardinality of maximal systems of rectangular islands is treated in [11]. The investigations on islands motivated further research on independence properties in lattices, see [3,4].…”

Islands: from coding theory to enumerative combinatorics and to lattice theory - overview and open problems

“…This proof can be read in [4], and provided several research directions in lattice theory, e.g. [5,6,14].…”

“…After the present paper was submitted, new lattice theoretical approach of the island topic came to existence, see Czédli [2]. This approach is related to [5] and to [14].…”

“…In [5] the authors showed that the CD-bases in a finite distributive lattice have the same number of elements, and conversely, if all finite lattices in a lattice variety have this property, then the variety must coincide with the variety of distributive lattices. From the proof of the main result of [5] it is clear that if we consider the maximum number of islands of arbitrary connected shape on the rectangular board, then it is bounded from above by the cardinality of a maximal chain of the powerset of the set of the cells, i.e. by m n:…”

Classification lattices are geometric for complete atomistic lattices

“…This problem is related to the application of concept lattices to one of the main problems in group technology (see [2,[6][7][8]). Actually, the problem is also related with the study of CD-bases of a lattice (see [9,10]), or to the investigation of the decomposition systems of a closure system (cf. [11,12]).…”