Forbidden configurations for distributive and modular ordered sets
type: Item
instance of: scholarly article
title: Forbidden configurations for distributive and modular ordered sets
zbMATH Open document ID: 0674.06003
DOI: 10.1007/BF00353659
author: Ivan Chajda
author: Jiří Rachůnek
published in: Order
publication date: 1989-01-01T00:00:00Z
review text: Let A be an ordered set, \(a,b\in A\). Denote by L(a,b) (U(a,b)) the set of all lower (upper, respectively) bounds of a, b. A is called distributive whenever \(L(U(a,b),c)=L(U(L(a,c),L(b,c))\) for all a,b,c\(\in A\). A is called modular if \(a\leq c\) implies \(L(U(a,b),c)=L(U(a,L(b,c)))\) for all a,b,c\(\in A\). It is clear that every distributive ordered set is modular and, moreover, if A is a lattice, then it is modular (distributive) iff it is modular (distributive, respectively) as an ordered set. The authors study the natural question whether modular ordered sets and distributive ordered sets can be characterized by some ``forbidden configurations'' similarly as in the case of lattices. The paper presents such configurations in the form of so-called strong subsets and LU subsets.
Mathematics Subject Classification ID: 06A06
zbMATH DE Number: 4103138
zbMATH Keywords: distributive ordered set
zbMATH Keywords: modular ordered sets
zbMATH Keywords: forbidden configurations
zbMATH Keywords: strong subsets
zbMATH Keywords: LU subsets
reviewed by: Jaromír Duda
MaRDI profile type: Publication
cites work: Posets and generalized lattices
cites work: Q3313909
cites work: Q3926627
full work available at URL: https://doi.org/10.1007/bf00353659
OpenAlex ID: W2091219652
description: scientific article; zbMATH DE number 4103138