Ebook: Applications of Category Theory to Fuzzy Subsets

This book has a fundamental relationship to the International Seminar on Fuzzy Set Theory held each September in Linz, Austria. First, this volume is an extended account of the eleventh Seminar of 1989. Second, and more importantly, it is the culmination of the tradition of the preceding ten Seminars. The purpose of the Linz Seminar, since its inception, was and is to foster the development of the mathematical aspects of fuzzy sets. In the earlier years, this was accomplished by bringing together for a week small grou ps of mathematicians in various fields in an intimate, focused environment which promoted much informal, critical discussion in addition to formal presentations. Beginning with the tenth Seminar, the intimate setting was retained, but each Seminar narrowed in theme; and participation was broadened to include both younger scholars within, and established mathematicians outside, the mathematical mainstream of fuzzy sets theory. Most of the material of this book was developed over the years in close association with the Seminar or influenced by what transpired at Linz. For much of the content, it played a crucial role in either stimulating this material or in providing feedback and the necessary screening of ideas. Thus we may fairly say that the book, and the eleventh Seminar to which it is directly related, are in many respects a culmination of the previous Seminars.

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Content:
Front Matter....Pages i-ix
Introduction....Pages 3-5
Front Matter....Pages 7-7
Classification of Extremal Subobjects of Algebras Over SM-SET....Pages 9-31
M-valued Sets and Sheaves over Integral Commutative CL-Monoids....Pages 33-72
The Logic of Unbalanced Subobjects in a Category with Two Closed Structures....Pages 73-105
Front Matter....Pages 107-107
Fuzzy Filter Functions and Convergence....Pages 109-136
Convenient Topological Constructs....Pages 137-151
A Topological Universe Extension of FTS ....Pages 153-176
Categorical Frameworks for Stone Representation Theories....Pages 177-231
Front Matter....Pages 233-233
Pointless Metric Spaces and Fuzzy Spaces....Pages 235-244
Fuzzy Unit Interval and Fuzzy Paths....Pages 245-256
Lattice Morphisms, Sobriety, and Urysohn Lemmas....Pages 257-274
The Topological Modification of the L-fuzzy Unit Interval....Pages 275-305
A Categorical Approach to Fuzzy Relational Database Theory....Pages 307-324
Fuzzy Points and Membership....Pages 325-334
Back Matter....Pages 335-399

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Content:
Front Matter....Pages i-ix
Introduction....Pages 3-5
Front Matter....Pages 7-7
Classification of Extremal Subobjects of Algebras Over SM-SET....Pages 9-31
M-valued Sets and Sheaves over Integral Commutative CL-Monoids....Pages 33-72
The Logic of Unbalanced Subobjects in a Category with Two Closed Structures....Pages 73-105
Front Matter....Pages 107-107
Fuzzy Filter Functions and Convergence....Pages 109-136
Convenient Topological Constructs....Pages 137-151
A Topological Universe Extension of FTS ....Pages 153-176
Categorical Frameworks for Stone Representation Theories....Pages 177-231
Front Matter....Pages 233-233
Pointless Metric Spaces and Fuzzy Spaces....Pages 235-244
Fuzzy Unit Interval and Fuzzy Paths....Pages 245-256
Lattice Morphisms, Sobriety, and Urysohn Lemmas....Pages 257-274
The Topological Modification of the L-fuzzy Unit Interval....Pages 275-305
A Categorical Approach to Fuzzy Relational Database Theory....Pages 307-324
Fuzzy Points and Membership....Pages 325-334
Back Matter....Pages 335-399
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