CDW-independent subsets in distributive lattices

A subset X of a lattice L with 0 is called CDW-independent if (1) it is CDindependent, i.e., for any x, y ∈ X , either x ≤ y or y ≤ x or x ∧ y = 0 and (2) it is weakly independent, i.e., for any n ∈ N and x, y1, . . . , yn ∈ X the inequalityx ≤ y1∨· · ·∨yn implies x ≤ yi for some i. A maximal CDW-independent subset is called a CDW-basis. With combinatorial examples and motivations in the background, the present paper points out that any two CDW-bases of a finite distributive lattice have the same number of elements. Moreover, if a lattice variety V contains a nondistributive lattice then there exists a finite lattice L in V such that L has CDW-bases X and Y with |X | 6= |Y |. 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 [2] and [3], and there was a decade witnessing an intensive study of weak independence, cf. Lengvárszky’s [8] and his other papers. Recently, the result of [2] has been successfully applied to combinatorial problems, cf. [4], Barát, Hajnal and Horváth [1], Horváth, Németh and Pluhár [6], and Pluhár [11]. Interestingly enough, many subsets occurring in these combinatorial applications [1], [4], [6] and [11], and also in E. K. Horváth, G. Horváth, Németh and Szabó [7], and Lengvárszky [9] and [10], enjoy another property, which has recently been named CD-independence in [5]. The present paper is motivated by the observation that a lot of subsets occurring in the combinatorial papers [1], [4], [6], [7], [9], [10] and [11] are both CD-independent and weakly independent. In fact, instead of [2], the main result of the present short note would also be applicable in most of these papers. Now, let us recall resp. introduce the basic definitions. Let L be a lattice with 0. A subset H of L is called weakly independent iff for all h, h1, . . .hn ∈ H which satisfy h ≤ h1 ∨ · · · ∨ hn there exists an i ∈ {1, . . . , n} such that h ≤ hi. 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. Subsets which are both CD-independent and weakly independent will be called CDW-independent subsets. Maximal CDW-independent subsets of Date: Submitted April 6, 2008, revised November 10, 2008.