Comparability graphs of lattices

A theorem of N. Terai and T. Hibi for finite distributive lattices and a theorem of Hibi for finite modular lattices (suggested by R.P. Stanley) are equivalent to the following: if a finite distributive or modular lattice of rank dd contains a complemented rank 3 interval, then the lattice is (d+1)(d+1)-connected.In this paper, the following generalization is proved: Let LL be a (finite or infinite) semimodular lattice of rank dd that is not a chain (d∈N0d∈N0). Then the comparability graph of LL is (d+1)(d+1)-connected if and only if LL has no simplicial elements, where z∈Lz∈L is simplicial if the elements comparable to zz form a chain.