On lattices with Möbius function \(\pm 1,0\)
type: Item
instance of: scholarly article
title: On lattices with Möbius function \(\pm 1,0\)
zbMATH Open document ID: 0611.06007
DOI: 10.1007/BF02187866
published in: Discrete \& Computational Geometry
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/131005
review text: 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.
Mathematics Subject Classification ID: 06B05
Mathematics Subject Classification ID: 52C07
Mathematics Subject Classification ID: 05A19
zbMATH DE Number: 3987341
zbMATH Keywords: Möbius functions
zbMATH Keywords: lattices
zbMATH Keywords: reliability
zbMATH Keywords: convex polytopes
zbMATH Keywords: evasiveness
author: J. Kahn
MaRDI profile type: Publication
cites work: Q3050437
cites work: Q4046079
cites work: Q3852212
cites work: A combinatorial perspective on the non-Radon partitions
cites work: The acyclic sets of an oriented matroid
cites work: A class of lattices with Möbius function \(\pm 1,0\)
cites work: A topological approach to evasiveness
cites work: Convexity in oriented matroids
cites work: Q3048571
cites work: On recognizing graph properties from adjacency matrices
cites work: On the foundations of combinatorial theory I. Theory of M�bius Functions
cites work: Q4053713
cites work: A Unified Formula for Analysis of Some Network Reliability Problems
cites work: Combinatorial properties of directed graphs useful in computing network reliability
cites work: New Topological Formula and Rapid Algorithm for Reliability Analysis of Complex Networks
cites work: A theorem concerning cyclic directed graphs with applications to network reliability
cites work: Euler's relation, Moebius functions and matroid identities
description: scientific article; zbMATH DE number 3987341