On lattices with Möbius function \(\pm 1,0\) (Q1087570)

This paper provides simple arguments of a geometric nature to explain why the Möbius functions of certain lattices take only the values -1, 0, 1. The lattices considered are all of the form \[ L_ 1(K)=\{\cup_{X\in M}X: M\subseteq K\}\quad or\quad L_ 2(K)=\{\emptyset \}\cup \{X: \exists Y\in K,\quad Y\subseteq X\}, \] where K is a collection of nonempty subsets of a finite set A. Results include a theorem of C. Greene on intervals from \(\{\) 1,...,n\(\}\) and theorems related to network reliability. The proofs exhibit connections with convex polytopes and the algorithmic notion of evasiveness.