Representation theorems for directed completions of consistent algebraic \(L\)-domains
type: Item
instance of: scholarly article
title: Representation theorems for directed completions of consistent algebraic \(L\)-domains
zbMATH Open document ID: 1096.06005
author: Xuxin Mao
author: Luoshan Xu
published in: Algebra Universalis
publication date: 2006-03-21T00:00:00Z
review text: The authors consider algebraic domains, in particular algebraic \(L\)-domains, with the weakened completeness condition that directed sets that are bounded above have a least upper bound, what they call consistent algebraic domains. They consider the dcpo obtained by taking the specialization order of the sobrification of the Scott topology, what they call the directed completion, and give various characterizations for it, e.g., it agrees with the ideal completion of the poset of compact elements. It is shown that eliminating a set of maximal elements with empty interior from an algebraic \(L\)-domain yields a consistent algebraic \(L\)-domain whose directed completion returns the original \(L\)-domain. They also show that cartesian closedness extends to the category of consistent algebraic \(L\)-domains.
Mathematics Subject Classification ID: 06B35
Mathematics Subject Classification ID: 54H10
Mathematics Subject Classification ID: 06A11
Mathematics Subject Classification ID: 18D15
zbMATH DE Number: 5013978
zbMATH Keywords: algebraic domain
zbMATH Keywords: directed completion
zbMATH Keywords: \(L\)-domain
zbMATH Keywords: Scott topology
zbMATH Keywords: sobrification
zbMATH Keywords: stable open set
reviewed by: Jimmie D. Lawson
MaRDI profile type: Publication
full work available at URL: s00012-005-1953-x
OpenAlex ID: W1970325809
DOI: 10.1007/S00012-005-1953-X
description: scientific article; zbMATH DE number 5013978