Matchings for linearly indecomposable modular lattices (Q1086266)

Let L be a finite lattice and let J(L) (M(L), resp.) denote the set of all nonzero join irreducible (all nonunit meet irreducible, resp.) elements of L. The author proves that every finite linearly indecomposable modular lattice L admits a bijection f: J(L)\(\to M(L)\) such that \(x\leq f(x)\) for any \(x\in J(L)\). (A lattice L is called linearly indecomposable if there do not exist x,y\(\in L\) with \(x<y\) such that \(z\leq x\) or \(z\geq y\) for all \(z\in L-\{x,y\}.)\) This result relates to a conjecture by \textit{I. Rival} [Algebra Univers. 6, 303-311 (1976; Zbl 0423.06008)] and complements the theorem of \textit{J. P. S. Kung} [Order 2, 105-112 (1985; Zbl 0582.06008)].