Stephen Ernest Rodabaugh

Categorical Foundations of Variable-Basis Fuzzy Topology

This chapter lays categorical foundations for topology and fuzzy topology in which the basis of a space—the lattice of membership values—is allowed to change from one object to another within the same category (the basis of a space being distinguished from the basis of the topology of a space). It is the goal of this chapter to create foundations which answer all the following questions in the affirmative:

Powerset Operator Foundations For Poslat Fuzzy Set Theories And Topologies

This chapter summarizes those powerset operator foundations of all mathematical and fuzzy set disciplines in which the operations of taking the image and preimage of (fuzzy) subsets play a fundamental role; such disciplines include algebra, measure theory and analysis, and topology. We first outline such foundations for the fixed-basis case—where the lattice of membership values or basis is fixed for objects in a particular category, and then extend these foundations to the variable-basis case—where the basis is allowed to vary from object to object within a particular category. Such foundations underlie almost all chapters of this volume. Additional applications include justifications for the Zadeh Extension Principle [19] and characterizations of fuzzy associative memories in the sense of Kosko [9]. Full proofs of all results, along with additional material, are found in Rodabaugh [16]—no proofs are repeated even when a result below extends its counterpart of [16]; some results are also found in Manes [11] and Rodabaugh [14, 15].

Point-set lattice-theoretic topology

This essay attempts to survey in a coherent way certain aspects of point-set lattice-theoretic or poslat topology, by which we mean (fuzzy) topology grounded in notions of sets, functions, powersets, and powerset operators - the latter two being lattice-theoretic in nature and examined from a lattice-theoretic point of view using methods of category theory. Connections with other related issues and developments are frequently given

Applications of category theory to fuzzy subsets

I: Topos-like and Model-Theoretic Approaches.- 1: Classification of Extremal Subobjects of Algebras over SM-SET.- 2: M-valued Sets and Sheaves over Integral Commutative CL-Monoids.- 3: The Logic of Unbalanced Subobjects in a Category with Two Closed Structures.- II: Categorical Methods in Topology.- 4: Fuzzy Filter Functors and Convergence.- 5: Convenient Topological Constructs.- 6: A Topological Universe Extension of FTS.- 7: Categorical Frameworks for Stone Representation Theories.- III: Applications and Related Topics in Logic and Topology.- 8: Pointless Metric Spaces and Fuzzy Spaces.- 9: Fuzzy Unit Interval and Fuzzy Paths.- 10: Lattice Morphisms, Sobriety, and Urysohn Lemmas.- 11: The Topological Modification of the L-Fuzzy Unit Interval.- 12: A Categorical Approach to Fuzzy Relational Database Theory.- 13: Fuzzy Points and Membership.- Appendices.- Index of Categories.- Addenda et Corrigenda.

Fuzzy addition in the L-fuzzy real line

Abstract Under the hypothesis L is a chain, we construct a binary operation ⊕ on the L-fuzzy real line R (L) which reduces to the usual addition on R if ⊕ is restricted to the embedded image of R in R (L), which yields a partially ordered, abelian cancellation semigroup with identity, and which is jointly fuzzy continuous on R (L). We show ⊕ is unique, i.e. it is the only extension of addition to R (L) which is consistent. We study the relationship between ⊕ and other fuzzy continuous extensions of the usual addition. We also show that fuzzy translation is a weak fuzzy homeomorphism and, under certain conditions, a fuzzy homeomorphism.

Relationship of Algebraic Theories to Powerset Theories and Fuzzy Topological Theories for Lattice-Valued Mathematics

This paper deals with a broad question—to what extent is topology algebraic—using two specific questions: (1) what are the algebraic conditions on the underlying membership lattices which insure that categories for topology and fuzzy topology are indeed topological categories; and (2) what are the algebraic conditions which insure that algebraic theories in the sense of Manes are a foundation for the powerset theories generating topological categories for topology and fuzzy topology? This paper answers the first question by generalizing the Hohle-Sostak foundations for fixed-basis lattice-valued topology and the Rodabaugh foundations for variable-basis lattice-valued topology using semi-quantales; and it answers the second question by giving necessary and sufficient conditions under which certain theories—the very ones generating powerset theories generating (fuzzy) topological theories in the sense of this paper—are algebraic theories, and these conditions use unital quantales. The algebraic conditions answering the second question are much stronger than those answering the first question. The syntactic benefits of having an algebraic theory as a foundation for the powerset theory underlying a (fuzzy) topological theory are explored; the relationship between these two specific questions is discussed; the role of pseudo-adjoints is identified in variable-basis powerset theories which are algebraically generated; the relationships between topological theories in the sense of Adamek-Herrlich-Strecker and topological theories in the sense of this paper are fully resolved; lower-image operators introduced for fixed-basis mathematics are completely described in terms of standard image operators; certain algebraic theories are given which determine powerset theories determining a new class of variable-basis categories for topology and fuzzy topology using new preimage operators; and the theories of this paper are undergirded throughout by several extensive inventories of examples.

A categorical accommodation of various notions of fuzzy topology

Different definitions of fuzzy topology have been stated and developed in the literature but have not been satisfactorily related to each other and to ordinary topology. To remedy this, a new fuzzy topological category FUZZ is defined: it is a significant generalization of previous definitions; it is the simplest setting yet constructed into which ordinary topology and these previous definitions may be placed (by identifying each with a different subcategory of FUZZ); it therefore allows us to conveniently analyze the relative merits of previous definitions, give coherence to known results, and indicate appropriate directions for future development; and it generates and may be viewed as both including and included in categories of fuzzy topological spaces which exhibit higher order fuzziness.

The Hausdorff separation axiom for fuzzy topological spaces

Abstract We develop a theory of α -Hausdorff fuzzy topological spaces which is compatible with α -compactness and fuzzy continuity, and for α a certain type of member of a given lattice we obtain characterizations of the α -Hausdorff subspaces of the fuzzy unit interval, the fuzzy open unit interval, and the fuzzy real line. In route we give an easy proof of the Fuzzy Tychonov Theorem for α -compactness and extend the theory of one-point α -compactifications.

Categorical Frameworks for Stone Representation Theories

The duality between topological spaces and lattices, first exploited by the famous Stone Representation Theorems [Stone 1936, 1937a, 1937b], is rightly regarded as one of the fundamental developments of twentieth century mathematics [Johnstone 1982]. The full expression of this duality in classical mathematics is seen in the relationship between topological spaces and frames/locales, culminating in the adjunction between topological spaces and locales [Papert-Papert 1958, Isbell 1972] and the resultant equivalence of sober topological spaces and spatial locales. For convenience, we dub this adjunction the Stone adjunction and the resulting (sometimes dual) categorical equivalences the Stone representation theory. The most complete account of these matters, and related issues such as the Stone-tech Compactification, may be found in [Johnstone 1982].

A theory of fuzzy uniformities with applications to the fuzzy real lines

Abstract For each completely distributive lattice L with order-reversing involution, the fuzzy real line R (L) is uniformizable by a uniformity which both generates the canonical (fuzzy) topology and induces a pseudometric generating the canonical topology. If L is also a chain, the usual addition and multiplication defined on R  R ({0, 1}) extend jointly (fuzzy) continuously to ⊕ and ⊙ on R (L). Three fundamental questions in fuzzy sets until now are: Question A . If L 1 ≅ L 2 , is R (L 1 ) uniformly isomorphic to R (L 2 ) in some sense? Question B . For each chain L , is ⊕ (jointly) uniformly continuous in a sense which guarantees its (joint) continuity on R (L)? Question C . Is R (L) a complete pseudometric space in some sense? We construct categories QU and U using the [quasi-] uniformities of B. Hutton which enable us to answer these questions in the affirmative. These results enhance the canonical standing of the fuzzy real lines and so give additional justification for answering in the affirmative: Question D . Does fuzzy topology have deep, specific, canonical examples?