Variable-basis topological systems versus variable-basis topological spaces
type: Item
instance of: scholarly article
title: Variable-basis topological systems versus variable-basis topological spaces
zbMATH Open document ID: 1197.54018
author: Sergejs Solovjovs
published in: Soft Computing
publication date: 2010-08-23T00:00:00Z
review text: Topological systems were introduced by [\textit{S. Vickers}, Topology via logic. Cambridge Tracts in Theoretical Computer Science, 5. Cambridge etc.: Cambridge University Press. (1989; Zbl 0668.54001)] as a framework in which to treat both point-sensitive spaces (i.e. topological spaces), the respective algebraic structures underlying their topologies (i.e. frames) and the corresponding point-free spaces (i.e. locales). On the other hand, there is the category of variable-basis lattice-valued topological spaces introduced by [\textit{S. E. Rodabaugh}, Categorical foundation of variable-basis fuzzy topology. Mathematics of fuzzy sets. Logic, topology, and measure theory. Dordrecht: Kluwer Academic Publishers. Handb. Fuzzy Sets Ser. 3, 273--388 (1999; Zbl 0968.54003)]. In the present paper, a variable-basis generalization of topological systems over an arbitrary variety of algebras is introduced and the functorial relationships between the categories of variable-basis topological systems and variable-basis lattice-valued topological spaces are investigated. This allows to treat variable-basis lattice-valued topological spaces (in the sense of Rodabaugh) and the respective algebraic structures underlying their topologies in a single framework. Some intrinsic properties (specifically products and coproducts) of the category of variable-basis topological systems are considered.
reviewed by: Jorge Picado
Mathematics Subject Classification ID: 54A40
Mathematics Subject Classification ID: 54B30
Mathematics Subject Classification ID: 18B30
Mathematics Subject Classification ID: 06D22
zbMATH DE Number: 5774066
zbMATH Keywords: fuzzy space
zbMATH Keywords: frame
zbMATH Keywords: locale
zbMATH Keywords: topological system
zbMATH Keywords: variable-basis topological system
zbMATH Keywords: lattice-valued topological space
zbMATH Keywords: variable-basis lattice-valued topological space
zbMATH Keywords: localic algebra
MaRDI profile type: Publication
full work available at URL: s00500-009-0485-2
OpenAlex ID: W2007010464
cites work: Quantales, observational logic and process semantics
cites work: Q4293501
cites work: Q3411966
cites work: Fuzzy topological spaces
cites work: Category theoretic properties of fuzzy topological spaces
cites work: A comparison of lattice-theoretic approaches to fuzzy topology
cites work: Q3747325
cites work: L-fuzzy sets
cites work: The fuzzy Tychonoff theorem
cites work: Q4440529
cites work: Q5430142
cites work: Fuzzy sets and sheaves. I: Basic concepts
cites work: Fuzzy sets and sheaves. II: Sheaf-theoretic foundations of fuzzy set theory with applications to algebra and topology
cites work: Products of fuzzy topological spaces
cites work: Atomless Parts of Spaces.
cites work: Q3965241
cites work: FUZZY TOPOLOGIES OF SCOTT CONTINUOUS FUNCTIONS AND THEIR RELATION TO THE HYPERGRAPH FUNCTOR
cites work: Q3717804
cites work: Fuzzy topological spaces and fuzzy compactness
cites work: Q4215784
cites work: Q4123496
cites work: Q5345553
cites work: Category theoretic aspects of chain-valued frames. I: Categorical and presheaf theoretic foundations
cites work: Category theoretic aspects of chain-valued frames. II: Applications to lattice-valued topology
cites work: Q4508560
cites work: Quantales, finite observations and strong bisimulation
cites work: A categorical accommodation of various notions of fuzzy topology
cites work: Point-set lattice-theoretic topology
cites work: POWERSET OPERATOR BASED FOUNDATION FORPOINT-SET LATTICE-THEORETIC (POSLAT) FUZZY SET THEORIES and TOPOLOGIES
cites work: Categorical foundation of variable-basis fuzzy topology
cites work: Relationship of algebraic theories to powerset theories and fuzzy topological theories for lattice-valued mathematics
cites work: Q5167548
cites work: On the category \(\mathcal {S}et(\mathcal {JCP}os)\)
cites work: Q3597368
cites work: Sobriety and spatiality in varieties of algebras
cites work: Q3994458
cites work: Fuzzy sets
DOI: 10.1007/S00500-009-0485-2
Recommended article: Interweaving algebra and topology: lattice-valued topological systems
Recommended article: Q4915515
Recommended article: Localification of variable-basis topological systems
Recommended article: Q3597368
Recommended article: Categorical foundations of variety-based bornology
Recommended article: Fuzzy algebras as a framework for fuzzy topology
Recommended article: Generalized fuzzy topology versus non-commutative topology
Recommended article: Lattice-valued preordered sets as lattice-valued topological systems
Recommended article: Q5496890
Recommended article: Pointed semi-quantales and lattice-valued topological spaces
description: scientific article; zbMATH DE number 5774066