CD-independent subsets in distributive lattices

The notion of CD-independence is introduced as follows. A subset X of a lattice L with 0 is called CD-independent if for any x, y ∈ 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 finite 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 less elements than a maximal chain, while lower semimodularity yields that each maximal chain together with all atoms form a CD-basis. Let L be a lattice with 0. A subset X of L will be called CD-independent if for any x, y ∈ X, either x ≤ y or y ≤ x or x∧y = 0. In other words, if any two elements of X either form a Chain (i.e., they are Comparable) or they are Disjoint; the initials explain our terminology. As one might expect, maximal CD-independent subsets are called CD-bases of L. The classical notion of independent subsets of (semimodular or modular) lattices has many applications ranging from von Neumann’s coordinatization theory to combinatorial aspects via matroid theory. Some other notions of independence were introduced in [1] and [2], and there was a decade witnessing an intensive study of weak independence, cf. Lengvárszky’s [9] and his other papers. Recently, the result of [1] has been successfully applied to combinatorial problems, cf. [3], Pluhár [10] and Horváth, Németh and Pluhár [8]. The present research started with the (easy) observation that many subsets occurring in [3], [8] and [10] are, in fact, CD-independent. There is a hope that CD-independence will have some applications in the future. Our lattices, usually denoted by L, are always assumed to be finite. To avoid technical inconvenience caused by trivial cases, in proofs and auxiliary notations we will often assume implicitly that |L| ≥ 3. As a general reference to (the rudiments of) lattice theory the reader is referred to Grätzer [6]. The set of all CD-bases of L will be denoted by B(L). For b ∈ L, ↓b will stand for the principal ideal {u ∈ L : u ≤ b}. The length, that is the supremum of {|C| − 1 : C is a chain in L}, of L is denoted by `(L). For u ∈ L, let h(u) = `(↓u) denote the height of u. If for all a, b, c ∈ L, a b implies a∨c b∨c then L is called semimodular. Lattices satisfying Date: February 29, 2008.