Convexity lattices
type: Item
instance of: scholarly article
title: Convexity lattices
zbMATH Open document ID: 0566.06005
DOI: 10.1007/BF01236802
published in: Algebra Universalis
publication date: 1985-01-01T00:00:00Z
review text: Generalizing the situation for vector spaces over ordered division rings the authors define the concept of convexity lattices namely of complete, biatomic, algebraic lattices with conditions that generalize Moore's four point theorem and the fact that in ordered sets of three points there is just one between the two others. Convexity lattices are called Peano if for distinct atoms p,q,r,s,t with \(q\leq p\bigvee r\), \(s\leq p\bigvee t\), there is an atom \(w\leq (r\bigvee s)\bigwedge (q\bigvee t)\). It is shown that they are Peano if and only if they satisfy the Pasch condition. In Peano convexity lattices the affine rank of an element can be defined, and some exchange property holds. The purpose of the article is to derive the theorems of Radon, Helly, and Carathéodory in the class of Peano convexity lattices. Especially it is shown that Helly's theorem holds in any atomic lattice which satisfies Radon's theorem.
reviewed by: Herbert Hotje
Mathematics Subject Classification ID: 06B05
Mathematics Subject Classification ID: 51G05
Mathematics Subject Classification ID: 52A35
Mathematics Subject Classification ID: 06B20
Mathematics Subject Classification ID: 51D05
zbMATH DE Number: 3902717
zbMATH Keywords: vector spaces over ordered division rings
zbMATH Keywords: convexity lattices
zbMATH Keywords: complete, biatomic, algebraic lattices
zbMATH Keywords: atoms
zbMATH Keywords: Pasch condition
zbMATH Keywords: Peano convexity lattices
zbMATH Keywords: exchange property
zbMATH Keywords: Helly's theorem
zbMATH Keywords: atomic lattice
zbMATH Keywords: Radon's theorem
author: Mary Katherine Bennett
author: Garrett. Birkhoff
MaRDI profile type: Publication
cites work: Q3724384
cites work: On generating affine geometries
cites work: Q5820602
cites work: Q3875978
cites work: On finite-modular atomistic lattices
cites work: Q5623714
full work available at URL: https://doi.org/10.1007/bf01236802
OpenAlex ID: W2328081108
description: scientific article; zbMATH DE number 3902717