On weakly independent subsets in lattices (Q1208079)

In ``Weakly independent subsets in lattices'' [ibid. 20, 194-196 (1985; Zbl 0569.06006)], \textit{G. Czedli}, \textit{A. P. Huhn} and \textit{E. T. Schmidt} proved that each element in a weakly independent set of a finite distributive lattice is a join-irreducible element in the sublattice generated by the set. In this short paper, the author proves that this result is true in any lattice. Moreover, it is clear that in a lattice \(L\) of finite length any weakly independent set has at most \(\ell(L)+1\) elements. The author gives a second property: let \(U\) be a weakly independent set in a lattice \(L\) of finite length such that \(| U|=\ell(L)+1\). Then the sublattice generated by \(U\) is distributive. We recall that a subset \(U\) of a lattice \(L\) is called weakly independent if whenever \(u,u_ 1,\dots,u_ n\) are elements of \(U\) such that \(u\leq u_ 1\lor\cdots\lor u_ n\) then \(u\leq u_ i\) for some \(1\leq i\leq n\).