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 d contains a complemented rank 3 interval, then the lattice is (d + 1)-connected. In this paper, the following generalization is proved: Let L be a (finite or infinite) semimodular lattice of rank d that is not a chain (d is an element of N-0). Then the comparability graph of L is (d + 1)-connected if and only if L has no simplicial elements, where z is an element of L is simplicial if the elements comparable to z form a chain. (c) 2007 Elsevier B.V. All rights reserved.