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 Title
 CATEGORICAL PROPERTIES OF LATTICEVALUED CONVERGENCE SPACES.
 Creator

Flores, Paul, Richardson, Gary, University of Central Florida
 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 LatticeValued 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...
Show moreThis work can be roughly divided into two parts. Initially, it may be considered a continuation of the very interesting research on the topic of LatticeValued 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 latticevalued 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 SLCS. Furthermore, connections between the diagonal axioms discussed and those given by Gähler are explored. In Chapter 4, further categorical properties of SLCS are discussed and in particular, it is shown that SLCS is topological, cartesian closed, and extensional. Chapter 5 explores connections between diagonal axioms for objects in the sub construct δ(PCS) and SLCS. Finally, recommendations for further research are provided.
Show less  Date Issued
 2007
 Identifier
 CFE0001715, ucf:47292
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFE0001715
 Title
 LatticeValued TFilters and Induced Structures.
 Creator

Reid, Frederick, Richardson, Gary, Brennan, Joseph, Han, Deguang, Lang, SheauDong, University of Central Florida
 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 Tconvergence structure which is defined in terms of Tfilters. This category is shown to be topological, strongly Cartesian closed, and extensional. It is well known that the category of...
Show moreA 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 Tconvergence structure which is defined in terms of Tfilters. 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 Tconvergence space has a compactification with the extension property provided the frame is a Boolean algebra. TCauchy spaces are defined and sufficient conditions for the existence of a completion are given. Tuniform limit spaces are also defined and their completions are given in terms of the TCauchy spaces they induce. Categorical properties of these subcategories are also investigated. Further, for a fixed Tconvergence 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 TCauchy spaces which induce the fixed space.
Show less  Date Issued
 2019
 Identifier
 CFE0007520, ucf:52586
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFE0007520