Annihilator characterizations of distributivity, modularity and semimodularity (Q1293151)

A graph \(G_L\) of a finite lattice \(L\) is the undirected Hasse diagram of \(L\). The distance \(d(a,b)\) between two elements \(a,b\) of \(L\) is their distance in \(G_L\). The shortest path between \(a\) and \(b\) in \(G_L\) is called an \(a\)-\(b\) geodesic. For any two elements \(a,b\) of \(L\) the geodesic annihilator \(\langle a,b\rangle_g\) is the set of all \(x\in L\) such that \(b\) is on an \(x\)-\(a\) geodesic in \(G_L\). By means of geodesic annihilators, characterizations of semimodular, modular and distributive lattices are presented. Further, conditions for a graph to be isomorphic to the graph of a semimodular, modular or distributive lattice are described.