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CATEGORICAL PROPERTIES OF LATTICE-VALUED CONVERGENCE SPACES

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Date Issued:
2007
Abstract/Description:
This work can be roughly divided into two parts. Initially, it may be considered a continuation of the very interesting research on the topic of Lattice-Valued Convergence Spaces given by Jäger [2001, 2005]. The alternate axioms presented here seem to lead to theorems having proofs more closely related to standard arguments used in Convergence Space theory when the Lattice is L=.Various Subcategories are investigated. One such subconstruct is shown to be isomorphic to the category of Lattice Valued Fuzzy Convergence Spaces defined and studied by Jäger . Our principal category is shown to be a topological universe and contains a subconstruct isomorphic to the category of probabilistic convergence spaces discussed in Kent and Richardson when L=. Fundamental work in lattice-valued convergence from the more general perspective of monads can be found in Gähler . Secondly, diagonal axioms are defined in the category whose objects consist of all the lattice valued convergence spaces. When the latter lattice is linearly ordered, a diagonal condition is given which characterizes those objects in the category that are determined by probabilistic convergence spaces which are topological. Certain background information regarding filters, convergence spaces, and diagonal axioms with its dual are given in Chapter 1. Chapter 2 describes Probabilistic Convergence and associated Diagonal axioms. Chapter 3 defines Jäger convergence and proves that Jäger's construct is isomorphic to a bireflective subconstruct of SL-CS. Furthermore, connections between the diagonal axioms discussed and those given by Gähler are explored. In Chapter 4, further categorical properties of SL-CS are discussed and in particular, it is shown that SL-CS is topological, cartesian closed, and extensional. Chapter 5 explores connections between diagonal axioms for objects in the sub construct δ(PCS) and SL-CS. Finally, recommendations for further research are provided.
Title: CATEGORICAL PROPERTIES OF LATTICE-VALUED CONVERGENCE SPACES.
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Name(s): Flores, Paul, Author
Richardson, Gary, Committee Chair
University of Central Florida, Degree Grantor
Type of Resource: text
Date Issued: 2007
Publisher: University of Central Florida
Language(s): English
Abstract/Description: This work can be roughly divided into two parts. Initially, it may be considered a continuation of the very interesting research on the topic of Lattice-Valued Convergence Spaces given by Jäger [2001, 2005]. The alternate axioms presented here seem to lead to theorems having proofs more closely related to standard arguments used in Convergence Space theory when the Lattice is L=.Various Subcategories are investigated. One such subconstruct is shown to be isomorphic to the category of Lattice Valued Fuzzy Convergence Spaces defined and studied by Jäger . Our principal category is shown to be a topological universe and contains a subconstruct isomorphic to the category of probabilistic convergence spaces discussed in Kent and Richardson when L=. Fundamental work in lattice-valued convergence from the more general perspective of monads can be found in Gähler . Secondly, diagonal axioms are defined in the category whose objects consist of all the lattice valued convergence spaces. When the latter lattice is linearly ordered, a diagonal condition is given which characterizes those objects in the category that are determined by probabilistic convergence spaces which are topological. Certain background information regarding filters, convergence spaces, and diagonal axioms with its dual are given in Chapter 1. Chapter 2 describes Probabilistic Convergence and associated Diagonal axioms. Chapter 3 defines Jäger convergence and proves that Jäger's construct is isomorphic to a bireflective subconstruct of SL-CS. Furthermore, connections between the diagonal axioms discussed and those given by Gähler are explored. In Chapter 4, further categorical properties of SL-CS are discussed and in particular, it is shown that SL-CS is topological, cartesian closed, and extensional. Chapter 5 explores connections between diagonal axioms for objects in the sub construct δ(PCS) and SL-CS. Finally, recommendations for further research are provided.
Identifier: CFE0001715 (IID), ucf:47292 (fedora)
Note(s): 2007-08-01
Ph.D.
Sciences, Department of Mathematics
Doctorate
This record was generated from author submitted information.
Subject(s): Topology
Stratified L-Filter
Lattice-Valued Convergence Structures
Categorical Properties
Persistent Link to This Record: http://purl.flvc.org/ucf/fd/CFE0001715
Restrictions on Access: public
Host Institution: UCF

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