Two notes on independent subsets in lattices (Q1124618)

A subset H of a lattice is called weakly independent (resp. *- independent) iff for all \(h,h_ 1,...,h_ n\in H\) satisfying \(h\leq h_ 1\vee...\vee h_ n\) (resp. \(h=h_ 1\vee...\vee h_ n)\) there is an \(i\in \{1,...,n\}\) such that \(h\leq h_ i\) (resp. \(h=h_ i)\). A maximal weakly independent (resp. *-independent) subset is called a weak basis (resp. *-basis) of L. Let \(J_ 0(L)\) denote the set of all join- irreducible elements of L. For an arbitrary finite lattice L, \(J_ 0(L)\) and all maximal chains are *-bases, and in case that the finite lattice L is distributive, these sets are also weak bases [\textit{G. Czédli}, \textit{A. P. Huhn} and \textit{E. T. Schmidt}, Algebra Univ. 20, 194-196 (1985; Zbl 0569.06006)]. It is well known that for every maximal chain C in a finite distributive lattice L, \(| J_ 0(L)| =| C|\). In the present paper, the authors prove the following generalization: Every *-basis of a finite distributive lattice L has at least \(| J_ 0(L)|\) elements, and every maximal chain in an arbitrary finite lattice L has at most \(| J_ 0(L)|\) elements. Furthermore, the authors show: If any two weak bases of a finite lattice L have the same cardinality, then L is modular. This is partially a converse of the following result in the cited paper: Any two weak bases of a finite distributive lattice have the same cardinality.