Abstract. A subset $X$ of a lattice $L$ with 0 is called {\it CDW-independent} if (1) it is CD-independent, i.e., for any $x,y\in X$, either $x\le y$ or $y\le x$ or $x\wedge y=0$ and (2) it is weakly independent, i.e., for any $n\in{\msbm N}$ and $x,y_1,\ldots, y_n\in X$ the inequality $x\le y_1\vee\cdots \vee y_n$ implies $x\le y_i$ 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 ${\cal V}$ contains a nondistributive lattice then there exists a finite lattice $L$ in ${\cal V}$ such that $L$ has CDW-bases $X$ and $Y$ with $|X|\not=|Y|$.

AMS Subject Classification (1991): 06D99

Keyword(s): lattice, distributivity, semimodularity, independent subset, CD-independent subset, weakly independent subset, CDW-independent subset, CDW-basis CD-basis

Received April 7, 2008, and in revised form November 10, 2008. (Registered under 6059/2009.)