CD-independent subsets in distributive lattices
Gábor Czédli, Miklós Hartmann, E. T. Schmidt · Publicationes Mathematicae Debrecen · 2009
A subset X of a lattice L with 0 is called CD-independent if for any x; y 2 X, either x · y or y · x or x^y = 0. In other words, if any two elements of X are either comparable or \disjoint. Maximal CD-independent subsets are called CD-bases. The main result says that any two CD-bases of a ¯nite distributive lattice L have the same number of elements. It is also shown that distributivity cannot be replaced by a weaker lattice identity. However, weaker assumptions on L are still relevant: semimodularity implies that no CD-basis can have fewer elements than a maximal chain, while lower semimodularity yields that each maximal chain together with all atoms forms a CD-basis.