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Lattice-Valued T-Filters and Induced Structures

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Date Issued:
2019
Abstract/Description:
A complete lattice is called a frame provided meets distribute over arbitrary joins. The implication operation in this context plays a central role. Intuitively, it measures the degree to which one element is less than or equal to another. In this setting, a category is defined by equipping each set with a T-convergence structure which is defined in terms of T-filters. This category is shown to be topological, strongly Cartesian closed, and extensional. It is well known that the category of topological spaces and continuous maps is neither Cartesian closed nor extensional.Subcategories of compact and of complete spaces are investigated. It is shown that each T-convergence space has a compactification with the extension property provided the frame is a Boolean algebra. T-Cauchy spaces are defined and sufficient conditions for the existence of a completion are given. T-uniform limit spaces are also defined and their completions are given in terms of the T-Cauchy spaces they induce. Categorical properties of these subcategories are also investigated. Further, for a fixed T-convergence space, under suitable conditions, it is shown that there exists an order preserving bijection between the set of all strict, regular, Hausdorff compactifications and the set of all totally bounded T-Cauchy spaces which induce the fixed space.
Title: Lattice-Valued T-Filters and Induced Structures.
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Name(s): Reid, Frederick, Author
Richardson, Gary, Committee Chair
Brennan, Joseph, Committee Member
Han, Deguang, Committee Member
Lang, Sheau-Dong, Committee Member
University of Central Florida, Degree Grantor
Type of Resource: text
Date Issued: 2019
Publisher: University of Central Florida
Language(s): English
Abstract/Description: A complete lattice is called a frame provided meets distribute over arbitrary joins. The implication operation in this context plays a central role. Intuitively, it measures the degree to which one element is less than or equal to another. In this setting, a category is defined by equipping each set with a T-convergence structure which is defined in terms of T-filters. This category is shown to be topological, strongly Cartesian closed, and extensional. It is well known that the category of topological spaces and continuous maps is neither Cartesian closed nor extensional.Subcategories of compact and of complete spaces are investigated. It is shown that each T-convergence space has a compactification with the extension property provided the frame is a Boolean algebra. T-Cauchy spaces are defined and sufficient conditions for the existence of a completion are given. T-uniform limit spaces are also defined and their completions are given in terms of the T-Cauchy spaces they induce. Categorical properties of these subcategories are also investigated. Further, for a fixed T-convergence space, under suitable conditions, it is shown that there exists an order preserving bijection between the set of all strict, regular, Hausdorff compactifications and the set of all totally bounded T-Cauchy spaces which induce the fixed space.
Identifier: CFE0007520 (IID), ucf:52586 (fedora)
Note(s): 2019-05-01
Ph.D.
Sciences, Mathematics
Doctoral
This record was generated from author submitted information.
Subject(s): Fuzzy Topology -- Lattice-valued Topology -- Fuzzy Sets -- L-Sets -- Lattice-valued Sets -- Fuzzy Filters -- Lattice-valued Filters -- T-Filters -- Fuzzy Ultrafilters -- T-Ultrafilters -- Lattice-valued Ultrafilters -- Fuzzy Closure -- Closure -- Regularity -- Topological -- Pretopological -- Compactification -- Completion -- Selection Maps -- Fuzzy Convergence Spaces -- T-Convergence Spaces -- Lattice-valued Convergence Spaces -- Fuzzy Cauchy Spaces -- T-Cauchy Spaces -- Lattice-valued Cauchy Spaces -- Fuzzy Limit Space -- T-Limit Spaces -- Lattice-valued Limit Spaces -- Fuzzy Uniform Limit Spaces -- T-Uniform Limit Spaces -- Lattice-valued Uniform Limit Spaces -- Stratified Convergence Spaces -- Stratified Filters -- Stratified Ultrafilters -- Strict Regular Harsdorf Embedding -- Category -- Strongly Cartesian Closed -- Extensional -- Topological Constructs -- Topological Universe -- Initial Structure -- Final Structure -- Reflective -- Bicoreflective
Persistent Link to This Record: http://purl.flvc.org/ucf/fd/CFE0007520
Restrictions on Access: public 2019-05-15
Host Institution: UCF

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