A lexicographic shellability characterization of geometric lattices (Q2434880)

In this interesting paper, the authors characterize geometric lattices. The following characterizations of geometric lattices are proved. Theorem. Let \(L\) be a finite atomic lattice. Then \(L\) is geometric if and only if every atom ordering induces a minimal labeling that is an \(EL\)-labeling. Theorem. A finite lattice \(L\) is a geometric lattice if and only if every ordering of the join-irreducibles induces a minimal labeling \(\lambda_{\gamma}\) which is an \(EL\)-labeling. The following theorem gives a condition, under which a finite lattice is semimodular. Theorem. Let \(L\) be a finite lattice with \(|\text{JoinIrred}(L)|=n\). Suppose that for every linear extension \(\gamma: \text{JoinIrred}(L)\to [n]\) of the subposet \(\text{JoinIrred}(L)\) of join-irreducibles in \(L\), the resulting minimal labeling \(\lambda_{\gamma}\) is an \(EL\)-labeling on \(L\). Then \(L\) is (upper) semimodular.