Spaces of type n on partially ordered sets
type: Item
instance of: scholarly article
title: Spaces of type n on partially ordered sets
zbMATH Open document ID: 0674.51002
DOI: 10.1007/BF02424316
author: Stefan E. Schmidt
published in: Geometriae Dedicata
publication date: 1989-01-01T00:00:00Z
review text: On any partially ordered set M one can define an incidence relation by xIy if and only if \(x\leq y\). If M possesses a subset E, the elements of which will be considered as points, such that M and E have a common lower bound, that any n (or less) elements of E have a least upper bound in M, and that each element of M occurs as such a bound, then the pair (M,E) is called a space of type n. The author presents some lattice theoretic characterizations of certain spaces of type n. In particular, he shows that a generalization of the geometric Veblen-Young axiom corresponds to a so-called n-generation property, which is a generalized version of the lattice theoretic ``Verbindungssatz''. He studies modularity and distributivity in the underlying lattices and, finally, discusses some specializations and examples.
Mathematics Subject Classification ID: 51A05
Mathematics Subject Classification ID: 06A06
Mathematics Subject Classification ID: 06B23
Mathematics Subject Classification ID: 05B35
zbMATH DE Number: 4103852
zbMATH Keywords: incidence structures on partially ordered sets
zbMATH Keywords: complete lattices
zbMATH Keywords: Veblen-Young axiom
reviewed by: Franz B. Kalhoff
MaRDI profile type: Publication
cites work: Varieties with modular subalgebra lattices
cites work: A Quick Proof for a One-Dimensional Version of Liapounoff's Theorem
cites work: Modular subalgebra lattices
cites work: Q3232305
cites work: Polynomial hyperforms
description: scientific article; zbMATH DE number 4103852