Relationship of algebraic theories to powerset theories and fuzzy topological theories for lattice-valued mathematics
type: Item
instance of: scholarly article
title: Relationship of algebraic theories to powerset theories and fuzzy topological theories for lattice-valued mathematics
zbMATH Open document ID: 1145.54004
DOI: 10.1155/2007/43645
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2008-06-03T00:00:00Z
full work available at URL: https://eudml.org/doc/54502
review text: Author's summary: This paper deals with a broad question -- to what extent is topology algebraic -- using two specific questions: (1) what are the algebraic conditions on the underlying membership lattices which insure that categories for topology and fuzzy topology are indeed topological categories; and (2) what are the algebraic conditions which insure that algebraic theories in the sense of Manes are a foundation for the powerset theories generating topological categories for topology and fuzzy topology? This paper answers the first question by generalizing the Höhle-Šostak foundations for fixed-basis lattice-valued topology and the Rodabaugh foundations for variable-basis lattice-valued topology using semi-quantales; and it answers the second question by giving necessary and sufficient conditions under which certain theories -- the very ones generating powerset theories generating (fuzzy) topological theories in the sense of this paper -- are algebraic theories, and these conditions use unital quantales. The algebraic conditions answering the second question are much stronger than those answering the first question. The syntactic benefits of having an algebraic theory as a foundation for the powerset theory underlying a (fuzzy) topological theory are explored; the relationship between these two specific questions is discussed; the role of pseudo-adjoints is identified in variable-basis powerset theories which are algebraically generated; the relationships between topological theories in the sense of Adámek-Herrlich-Strecker and topological theories in the sense of this paper are fully resolved; lower-image operators introduced for fixed-basis mathematics are completely described in terms of standard image operators; certain algebraic theories are given which determine powerset theories determining a new class of variable-basis categories for topology and fuzzy topology using new preimage operators; and the theories of this paper are undergirded throughout by sev…
reviewed by: Bernhard Behrens
Mathematics Subject Classification ID: 54A40
Mathematics Subject Classification ID: 54B30
Mathematics Subject Classification ID: 06F07
Mathematics Subject Classification ID: 18B99
Mathematics Subject Classification ID: 18C10
zbMATH DE Number: 5282468
zbMATH Keywords: quantale
zbMATH Keywords: fixed-based (variable basis) lattice-valued topology
zbMATH Keywords: topological category
zbMATH Keywords: algebraic theory
zbMATH Keywords: powerset theory
zbMATH Keywords: algebraic generation of (topological) powerset theories
author: S. E. Rodabaugh
MaRDI profile type: Publication
OpenAlex ID: W2055390537
cites work: Axiomatic foundations of fixed-basis fuzzy topology
cites work: Categorical foundation of variable-basis fuzzy topology
cites work: Q4123496
cites work: Relationship of algebraic theories to powersets over objects in \(\mathbf {Set}\) and \(\mathbf {Set}\times C\)
cites work: Q4693062
cites work: Q4037041
cites work: Some remarks on fuzzy powerset operators
cites work: Fuzzy topological properties and hereditariness.
cites work: Structured lattices and ground categories of \(L\)-sets
cites work: THE SUBSPACE PROBLEM IN THE TRADITIONAL POINT-SET CONTEXT OF FUZZY TOPOLOGY
cites work: Q4440529
cites work: Many valued topology and its applications
cites work: Q5532570
cites work: Q4836222
cites work: Q3996704
cites work: Q3965241
cites work: Triangular norms
cites work: POWERSET OPERATOR BASED FOUNDATION FORPOINT-SET LATTICE-THEORETIC (POSLAT) FUZZY SET THEORIES and TOPOLOGIES
cites work: Powerset operator foundations for poslat fuzzy set theories and topologies
cites work: Q4440521
cites work: Q4748790
cites work: A categorical accommodation of various notions of fuzzy topology
cites work: Point-set lattice-theoretic topology
cites work: Q4010279
cites work: Q4440524
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: Separation axioms: Representation theorems, compactness, and compactifications
cites work: Fuzzy topological spaces
cites work: The fuzzy Tychonoff theorem
cites work: Upper semicontinuous fuzzy sets and applications
cites work: A general theory of fuzzy topological spaces
cites work: Q4440530
cites work: Q4440519
cites work: LOWER SET-VALUED FUZZY TOPOLOGIES
cites work: On the category \(\mathcal {S}et(\mathcal {JCP}os)\)
cites work: Q3779407
cites work: Categories of lattice-valued sets as categories of arrows
cites work: Q4735749
cites work: Q3779408
cites work: Two decades of fuzzy topology: basic ideas, notions, and results
cites work: Q3485582
cites work: Fuzzy sets
cites work: Q4440526
cites work: Q4490110
cites work: Topological and algebraic structures in fuzzy sets. A handbook of recent developments in the mathematics of fuzzy sets
cites work: Applications of category theory to fuzzy subsets. Based on the eleventh international seminar on fuzzy set theory, held in Linz, Austria, September 1989
Recommended article: Relationship of algebraic theories to powersets over objects in \(\mathbf {Set}\) and \(\mathbf {Set}\times C\)
Recommended article: On the lattice and the algebra of fuzzy subsets of a universal algebra
Recommended article: Q4522377
Recommended article: FUZZY RELATIONS AND ALEXANDROV L-TOPOLOGIES
Recommended article: Q4405710
Recommended article: POWERSET OPERATOR BASED FOUNDATION FORPOINT-SET LATTICE-THEORETIC (POSLAT) FUZZY SET THEORIES and TOPOLOGIES

Relationship of algebraic theories to powerset theories and fuzzy topological theories for lattice-valued mathematics (Recommended article)
Recommended article: Q3496386
Similarity Score: 0.8686551
Recommender Run: Recommender Run 3

Relationship of algebraic theories to powerset theories and fuzzy topological theories for lattice-valued mathematics (Recommended article)
Recommended article: Q3712467
Similarity Score: 0.86668897
Recommender Run: Recommender Run 3

Relationship of algebraic theories to powerset theories and fuzzy topological theories for lattice-valued mathematics (Recommended article)
Recommended article: Powerset operator foundations for poslat fuzzy set theories and topologies
Similarity Score: 0.86462665
Recommender Run: Recommender Run 3

Relationship of algebraic theories to powerset theories and fuzzy topological theories for lattice-valued mathematics (Recommended article)
Recommended article: Representation theorem of fuzzy lattices and its applications in fuzzy topology
Similarity Score: 0.86251557
Recommender Run: Recommender Run 3