An Overview of the Fuzzy Axiomatic Systems and Characterizations Proposed at Ghent University
---
paper_title: Topological Spaces

paper_content:

Publisher Summary ::: The problem of defining a notion of convergence appropriate to any set whatsoever, a notion that lends itself to easy rules of calculation (like the rules for calculating with limits in the classical theory), leads almost inescapably to the concept of topological space. This chapter studies topological spaces from this viewpoint of the problem. The chapter presents a simple and appropriate transcription of specific properties of classical real analysis to define the general concept of topological space. The concept of topological spaces is derivable from a few extra conditions, imposable in a natural way on any notion of convergence. Moreover, it is proved that the second defining property of a topology is an appropriate generalization of the diagonal property of real sequences. Any concept of convergence that verifies conditions (α)–(γ), and in which the generalized diagonal property holds, can be thought of as having been generated by a topology.
---
paper_title: Lattice properties of neighbourhood systems in Chang fuzzy topological spaces

paper_content:

Abstract This paper is a natural continuation of a previous one concerning the different notions of point fuzzification and the corresponding neighbourhood structure. The approaches of Kerre, Pu-Liu and Mashhour are linked together and are looked at more deeply from a lattice-theoretical point of view. The gap between Lowen spaces and Chang spaces is partly filled.
---
paper_title: Fuzzy Sierpinski space and its generalizations

paper_content:

the Sierpinski space. It seemed reasonable to us that similar fuzzy topological spaces will be needed in the further development of fuzzy topolo,v. The key object in our construction of fuzzy topological spaces such as the Sierpinski space, the included (excluded) fuzzy singleton topology, and the included (excluded) fuzzy set topology, is the fuzzy singleton as described by Goguen. We have used fuzzy singletons instead of Wong’s fuzzy points because crisp fuzzy singletons reduce to ordinary ones and because we believe that every fuzzy generalization should be formulated in such a way that it contains the corresponding ordinary-set-theoretic notion as a special (crisp) case. The
---
paper_title: Continuity of fuzzy multivalued mappings

paper_content:

Continuity of multivalued mappings is characterized by two types of semi-continuity: lower and upper semi-continuity. In this paper, different equivalent ways in which these concepts can be expressed are investigated and then used to define lower and upper semi-continuous fuzzy multivalued mappings. Some shortcomings in existing definitions of upper semi-continuous multivalued mappings are addressed. Several relationships between the lower semi-continuity of fuzzy multivalued mappings and the classical lower semi-continuity of their cuts, or the lower semi-continuity w.r.t. the level topologies are established in the setting of stratified fuzzy topological spaces. Two different types of upper semi-continuity of fuzzy multivalued mappings are introduced, inspired by the two equivalent ones in the crisp case. The relationships between these two types are studied completely. Many interesting properties of lower and upper semi-continuous fuzzy multivalued mappings are presented, including the composition and union of lower and upper semi-continuous fuzzy multivalued mappings and the compactness preserving property of upper semi-continuous and compact-valued fuzzy multivalued mappings.
---
paper_title: A general treatment of data redundancy in a fuzzy relational data model

paper_content:

In fuzzy data modeling, there is a series of design issues that should be dealt with. This article concentrates on the definition of data redundancy in a fuzzy relational data model. Compared to the existing approaches, a general treatment is provided, enabling us to directly and rigourously handle the situation in which possibility distributions can appear as attribute values combined with closeness relations in domain elements. The treatment of fuzzy data redundancy, which we propose here, is mainly based on Zadeh's extension principle. In doing so, a natural incorporation of closeness relations into the operations defined by the extension principle is worked out. Certain desirable properties are achieved in our model, in terms of intuition and model design. © 1992 John Wiley & Sons, Inc.
---
paper_title: The dependency-preserving decomposition and a testing algorithm in a fuzzy relational data model

paper_content:

Abstract This paper focuses on one of the fundamental design issues in fuzzy database modeling, namely, the dependency-preserving decomposition. In a fuzzy relational data model where imprecision is reflected by possibility distributions for attribute values as well as by closeness relations for domain elements, a “poor” model design can be remedied, in many cases, by decomposing relation schemes in order to eliminate/reduce data redundancy and update anomalies. On the other hand, the decomposition should guarantee that the semantic knowledge and integrity constraints expressed by fuzzy functional dependency (FFD) are satisfactorily preserved by the resultant relation schemes. Based on the concept of FFD transitive closure, an algorithm has been developed to test whether a given decomposition is dependency-preserving with respect to a given set of FFDs. Finally, two special FFD sets, one composed of a X 1 -to- X 1 FFD loop and the other composed of a X 1 - X m FFD and a X 1 -to- X m FFD chain, are investigated.
---
paper_title: A COMPUTATIONAL ALGORITHM FOR THE FFD TRANSITIVE CLOSURE AND A COMPLETE AXIOMATIZATION OF FUZZY FUNCTIONAL DEPENDENCE (FFD).

paper_content:

This article focuses on the axiomatization of fuzzy functional dependency (FFD) in a fuzzy relational data model, which is considered as a fundamental issue towards building a theory of fuzzy relational database design. As an inference basis, the FFD axiomatic system composed of four FFD inference rules is established, and proven to be correct in terms of FFD logical implication with respect to the relation scheme. Consequently, the concept of the FFD transitive closure is defined, and correspondingly a computational algorithm is developed. This helps to tell whether a specific FFD can be derived from a given FFD set using those inference rules. In addition, some details of the algorithm as well as the aspects of the computational complexity and correctness are investigated. Finally, the soundness and completeness of the axiomatic system are discussed. It is proven that the system is both sound and complete. Importantly, this result enables us to “equate” the concept of FFD logical implication to the concept of FFD derivation using the inference rules. © 1994 John Wiley & Sons, Inc.
---
paper_title: Mathematics of Fuzziness―Basic Issues

paper_content:

Preliminaries.- Basics of Fuzzy Sets.- Fuzzy Relations.- Extension Principle and Fuzzy Numbers.- A Brief Introduction to Some Pure Mathematical Topics.- Fuzzy Inference and Fuzzy Control.
---
paper_title: A MODEL FOR THE MEASUREMENT OF MEMBERSHIP AND THE CONSEQUENCES OF ITS EMPIRICAL IMPLEMENTATION

paper_content:

Representation and uniqueness theorems are presented for the fundamental measurement of the membership of a fuzzy set, when the domain of discourse is order-dense. The conclusion that membership is on an interval scale is justified by the inapplicability of extensive measurement to fuzziness and the lack of a natural origin for membership. Some descriptive statistics are also presented from an empirical study currently in progress to construct membership functions. They suggest that subjects differ in their perception of fuzziness and that the models in the literature for certain linguistic operators such as ‘not’, ‘very’, and ‘antonym’ are not sufficiently general to account for the range of individual perceptions of vague attributes. A more general mathematical model is proposed. The question of the meaningfulness of statements about membership functions is also discussed, in the light of the assertion that membership can be fundamentally measured at best on an interval scale. Some of the difficulties which arise in fuzzy set theory due to this property are illustrated and a solution is suggested, involving the use of a function which is derived from membership but is on an absolute scale.
---
paper_title: Smets-Magrez axioms for R-implicators in interval-valued and intuitionistic fuzzy set theory

paper_content:

Interval-valued fuzzy sets constitute an extension of fuzzy sets which give an interval approximating the "real" (but unknown) membership degree. Interval-valued fuzzy sets are equivalent to intuitionistic fuzzy sets in the sense of Atanassov which give both a membership degree and a non-membership degree, whose sum must be smaller than or equal to 1. Both are equivalent to L-fuzzy sets w.r.t, a special lattice L*. In fuzzy set theory, an important class of triangular norms is the class of those that satisfy the residuation principle. In a previous paper5 we gave a construction for t-norms on L* satisfying the residuation principle which are not t-representable. In this paper we investigate the Smets-Magrez axioms and some other properties for the residual implicator generated by such t-norms.
---
paper_title: Implication in fuzzy logic

paper_content:

Abstract Intermediate truth values and the order relation “as true as” are interpreted. The material implication A → B quantifies the degree by which “B is at least as true as A.” Axioms for the → operator lead to a representation of → by the pseudo-Lukasiewicz model. A canonical scale for the truth value of a fuzzy proposition is selected such that the → operator is the Lukasiewicz operator and the negation is the classical 1−. operator. The mathematical structure of some conjunction and disjunction operators related to → are derived.
---
paper_title: On dependencies and independencies of fuzzy implication axioms

paper_content:

A fuzzy implication, commonly defined as a two-place operation on the unit interval, is an extension of the classical binary implication. It plays important roles in both mathematical and applied sides of fuzzy set theory. Besides the basic axioms, there are many potential fuzzy implication axioms, among which eight are widely used in the literature. Different fuzzy implications satisfying different subgroups of these eight axioms can be found. However, certain interrelationships exist between these eight axioms. But the results remain incomplete. This paper aims to lay bare the interrelationships between these eight axioms. The result is instrumental to propose a classification of fuzzy implications.
---

Title: An Overview of the Fuzzy Axiomatic Systems and Characterizations Proposed at Ghent University

Section 1: Overview
Description 1: Summarize the overall findings and structure of the paper, including brief descriptions of each section covered.

Section 2: On the Characterization of a Chang Fuzzy Topology by Means of Preassigned Operations
Description 2: Discuss the characterization of Chang fuzzy topologies using preassigned operations, such as the interior operator.

Section 3: On the Separation Axioms in Chang Fuzzy Topological Spaces
Description 3: Elaborate on the separation axioms within Chang fuzzy topological spaces and the dependencies among them.

Section 4: Axiomatics in Fuzzy Relational Databases
Description 4: Explain the axiomatization in the context of fuzzy relational databases, focusing on fuzzy functional dependencies.

Section 5: On the Characterization of a Fuzzy Preference Structure
Description 5: Describe the characteristics of fuzzy preference structures and their importance in modeling decision-making problems.

Section 6: On the Equivalency between L-fuzzy Sets and L-flou Sets
Description 6: Detail the equivalence between L-fuzzy sets and L-flou sets, and investigate the necessary conditions in the lattices.

Section 7: Axioms for the Ordering of Fuzzy Quantities
Description 7: Present the axioms introduced for the purpose of ordering fuzzy quantities and compare various ordering methods.

Section 8: Axioms for Defuzzification
Description 8: Outline the different techniques for defuzzification along with the proposed axioms that ensure their applicability and robustness.

Section 9: Axioms for a Fuzzy Implication
Description 9: Summarize the basic and additional axioms for fuzzy implications, including their significance in fuzzy logic and reasoning.

Section 10: Triangle Algebras: A Characterization of Interval-valued Residuated Lattices
Description 10: Describe the structure and properties of triangle algebras, focusing on interval-valued residuated lattices.