Weakly independent subsets in lattices
A subset H of a lattice L is weakly independent if for all \(h,h_ 1,h_ 2,...,h_ n\in H\) which satisfy \(h\leq h_ 1\vee h_ 2\vee...\vee h_ n\) there must exist an index i (1\(\leq i\leq n)\) such that \(h\leq h_ i\). A basis is a maximal weakly independent set. Let \(J_ 0(L)\) denote the set of join irreducible elements of the lattice L. The following are the main results of the paper. Theorem 1. Any two bases of a finite distributive lattice have the same cardinality. Theorem 2. A subset H of a finite distributive lattice L is a basis for L if and only if the sublattice K generated by H is a sublattice of maximal length and \(H=J_ 0(K)\). The paper closes with an example of a modular lattice for which both Theorems fail.
scientific article; zbMATH DE number 3908492