A CHARACTERIZATION OF SEMIMODULAR LATTICES WHOSE COMPARABILITY GRAPHS ARE WEAKLY CONNECTED

JONATHAN DAVID FARLEY AND STEFAN E. SCHMIDT

Abstract. Let L be a semimodular lattice of rank d that is not a chain (d ∈ ℕ_{0}). It is shown that the comparability graph of L is (d + 1)-connected if and only if L has no simplicial elements (where z ∈ L is simplicial if the elements comparable to z form a chain).

A theorem of T. Hibi and N. Terai for finite distributive lattices, and a theorem of Hibi for finite modular lattices (suggested by R. P. Stanley), are obtained as corollaries: To wit, if a finite modular or distributive lattice of rank d contains a complemented rank 3 interval, then the lattice is (d + 1)-connected.

The following structural results are obtained: If L is a rank d semimodular lattice and L \ C is disconnected, where C ⊆ L has at most d elements, then C is a d-element chain. Moreover, all but at most one of the components of L \ C are bottom-rooted trees whose leaves are simplicial elements. If L is modular, and a and b minimal elements from different components of L \ C, then either a or b is simplicial. Further, either (1) C is the set of elements greater or less than a simplicial element, or (2) L \ C is the disjoint union of a top-rooted tree and a bottom-rooted tree. If L is distributive, then either (1) holds or (2’) L is the ordinal sum of a chain, a product of two chains, and another chain. It is proved that a distributive lattice of rank d, with poset of join-irreducibles P, is (d+1)-connected if and only if P contains a three-element antichain or a four-element crown (provided L is not a chain).