On the covering graph of balanced lattices (Q1923532)

The paper concerns covering graphs of lattices, i.e. their undirected Hasse diagrams. By \(J(L)\) the set of join-irreducible non-zero elements of a lattice \(L\) is denoted, the symbol \(j'\) for \(j\in L\) denotes the unique lower cover of \(j\). A lattice \(L\) of finite length is called strong, if for all \(j\in J(L)\) and \(x\in L\) the inequality \(j\leq x\vee j'\) implies \(j\leq x\). If both \(L\) and its dual lattice are strong, the lattice \(L\) is called balanced. The main result is the following theorem: Let \(L\) and \(L'\) be graded lattices with graph isomorphic covering graphs. \(L\) is balanced if and only if \(L'\) is balanced. Moreover, if this condition is satisfied, then there are sublattices \(A\) and \(B\) of \(L\) such that \(L\cong A\times B\) and \(L'\cong A^d\times B\). (Here \(A^d\) denotes the lattice dual to \(A\)). There are two corollaries of this theorem, one of them was proved as a theorem by D. Duffus and I. Rival, the other by J. Jakubík.