Boolean and distributive ordered sets: Characterization and representation by sets
type: Item
instance of: scholarly article
title: Boolean and distributive ordered sets: Characterization and representation by sets
zbMATH Open document ID: 0838.06004
DOI: 10.1007/BF01108627
published in: Order
publication date: 1996-06-06T00:00:00Z
review text: Let \((A,\leq)\) be an ordered set and \(X\subseteq A\). Denote by \(L(X)\) the lower cone of \(X\) in \(A\) and by \(U(X)\) the upper cone of \(X\). An ordered set \((A, \leq)\) is said to be distributive if \(L(x, U(y,z))= L(U(L (x,y),L(x,z)))\) for all \(x,y,z\in A\). An element \(y\in A\) is called a complement of \(x\) in \((A,\leq)\) if \(L(x,y)= L(A)\) and \(U(x,y)= U(A)\). A Boolean ordered set is defined as a distributive ordered set of which each element has a complement in \(A\). Some well-known theorems such as the Glivenko-Stone theorem and the Stone representation theorem on Boolean algebras and the Birkhoff representation theorem for distributive lattices are generalized to Boolean ordered sets and to distributive ordered sets, respectively.
Mathematics Subject Classification ID: 06A06
zbMATH DE Number: 828190
zbMATH Keywords: prime ideal
zbMATH Keywords: maximal ideal
zbMATH Keywords: Boolean ordered set
zbMATH Keywords: distributive ordered set
author: Josef Niederle
reviewed by: V. N. Saliĭ
MaRDI profile type: Publication
cites work: Q3138111
cites work: Forbidden configurations for distributive and modular ordered sets
cites work: Ideals in Partially Ordered Sets
cites work: Q4298406
cites work: Q3035342
cites work: Q3982193
description: scientific article; zbMATH DE number 828190