A Characterization of Semimodular Lattices Whose Comparability Graphs Are Weakly Connected
Jonathan David Farley and Stefan E. Schmidt
Let $L$ be a semimodular lattice of rank $d$ that is not a chain ($d\in\Bbb N_0$). It is shown that the comparability graph of $L$ is $(d+1)$-connected if and only if $L$ has no simplicial elements (where $z\in L$ 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 $L\setminus C$ is disconnected, where $C\subseteq L$ has at most $d$ elements, then $C$ is a $d$-element chain. Moreover, all but at most one of the components of $L\setminus C$ are bottom-rooted trees whose leaves are simplicial elements. If $L$ is modular, and $a$ and $b$ minimal elements from different components of $L\setminus C$, 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) $L\setminus C$ 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).