\(Z\)-semicontinuous posets
type: Item
instance of: scholarly article
title: \(Z\)-semicontinuous posets
zbMATH Open document ID: 1061.06017
author: Robert C. Powers
author: Thomas Riedel
published in: Order
publication date: 2005-04-07T00:00:00Z
review text: A basic result in the theory of continuous lattices is that in a distributive continuous lattice, the way-below relation is multiplicative if and only if every weak prime element is prime if and only if the set of weak primes is closed in the Lawson topology. The authors generalize these results to the case of \(Z\)-semicontinuous posets, which include semicontinuous lattices and \(Z\)-continuous posets as special cases. For \(Z\)-prime and \(Z\)-semicontinuous posets, \(\ll_Z\) is multiplicative if and only if every weak prime is prime, and in the case that the poset \(P\) is further assumed to be a complete lattice, the set of weak primes is closed in a modified version of the Lawson topology \(\lambda_Z(P)\).
Mathematics Subject Classification ID: 06B35
Mathematics Subject Classification ID: 06A11
Mathematics Subject Classification ID: 06B30
zbMATH DE Number: 2153980
zbMATH Keywords: continuous lattice
zbMATH Keywords: \(Z\)-continuous poset
zbMATH Keywords: \(Z\)-semicontinuous poset
zbMATH Keywords: multiplicative relation
reviewed by: Jimmie D. Lawson
MaRDI profile type: Publication
full work available at URL: b:orde.0000034615.95682.7c
OpenAlex ID: W2031384256
DOI: 10.1023/B:ORDE.0000034615.95682.7C
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description: scientific article; zbMATH DE number 2153980