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LATTICE-VALUED CONVERGENCE: QUOTIENT MAPS

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Date Issued:
2008
Abstract/Description:
The introduction of fuzzy sets by Zadeh has created new research directions in many fields of mathematics. Fuzzy set theory was originally restricted to the lattice , but the thrust of more recent research has pertained to general lattices. The present work is primarily focused on the theory of lattice-valued convergence spaces; the category of lattice-valued convergence spaces has been shown to possess the following desirable categorical properties: topological, cartesian-closed, and extensional. Properties of quotient maps between objects in this category are investigated in this work; in particular, one of our principal results shows that quotient maps are productive under arbitrary products. A category of lattice-valued interior operators is defined and studied as well. Axioms are given in order for this category to be isomorphic to the category whose objects consist of all the stratified, lattice-valued, pretopological convergence spaces. Adding a lattice-valued convergence structure to a group leads to the creation of a new category whose objects are called lattice-valued convergence groups, and whose morphisms are all the continuous homomorphisms between objects. The latter category is studied and results related to separation properties are obtained. For the special lattice , continuous actions of a convergence semigroup on convergence spaces are investigated; in particular, invariance properties of actions as well as properties of a generalized quotient space are presented.
Title: LATTICE-VALUED CONVERGENCE: QUOTIENT MAPS.
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Name(s): Boustique, Hatim, Author
Richardson, Gary, Committee Chair
University of Central Florida, Degree Grantor
Type of Resource: text
Date Issued: 2008
Publisher: University of Central Florida
Language(s): English
Abstract/Description: The introduction of fuzzy sets by Zadeh has created new research directions in many fields of mathematics. Fuzzy set theory was originally restricted to the lattice , but the thrust of more recent research has pertained to general lattices. The present work is primarily focused on the theory of lattice-valued convergence spaces; the category of lattice-valued convergence spaces has been shown to possess the following desirable categorical properties: topological, cartesian-closed, and extensional. Properties of quotient maps between objects in this category are investigated in this work; in particular, one of our principal results shows that quotient maps are productive under arbitrary products. A category of lattice-valued interior operators is defined and studied as well. Axioms are given in order for this category to be isomorphic to the category whose objects consist of all the stratified, lattice-valued, pretopological convergence spaces. Adding a lattice-valued convergence structure to a group leads to the creation of a new category whose objects are called lattice-valued convergence groups, and whose morphisms are all the continuous homomorphisms between objects. The latter category is studied and results related to separation properties are obtained. For the special lattice , continuous actions of a convergence semigroup on convergence spaces are investigated; in particular, invariance properties of actions as well as properties of a generalized quotient space are presented.
Identifier: CFE0002369 (IID), ucf:47811 (fedora)
Note(s): 2008-12-01
Ph.D.
Sciences, Department of Mathematics
Doctorate
This record was generated from author submitted information.
Subject(s): Convergence Spaces
Lattice-Valued Convergence
Quotient Maps
Semigroup Actions
Persistent Link to This Record: http://purl.flvc.org/ucf/fd/CFE0002369
Restrictions on Access: public
Host Institution: UCF

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