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LATTICEVALUED CONVERGENCE: QUOTIENT MAPS
 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 latticevalued convergence spaces; the category of latticevalued convergence spaces has been shown to possess the following desirable categorical properties: topological, cartesianclosed, 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 latticevalued 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, latticevalued, pretopological convergence spaces. Adding a latticevalued convergence structure to a group leads to the creation of a new category whose objects are called latticevalued 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:  LATTICEVALUED 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 latticevalued convergence spaces; the category of latticevalued convergence spaces has been shown to possess the following desirable categorical properties: topological, cartesianclosed, 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 latticevalued 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, latticevalued, pretopological convergence spaces. Adding a latticevalued convergence structure to a group leads to the creation of a new category whose objects are called latticevalued 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): 
20081201 Ph.D. Sciences, Department of Mathematics Doctorate This record was generated from author submitted information. 

Subject(s): 
Convergence Spaces LatticeValued Convergence Quotient Maps Semigroup Actions 

Persistent Link to This Record:  http://purl.flvc.org/ucf/fd/CFE0002369  
Restrictions on Access:  public  
Host Institution:  UCF 