Roy Goetschel

Elementary fuzzy calculus

Abstract In [2] Dubois and Prade introduced the notion of fuzzy numbers and defined the basic operations of addition, subtraction, multiplication, and division. A slightly modified definition of fuzzy numbers was presented in [4], and in that paper a metric was defined for this family of fuzzy sets. Another less restrictive definition of fuzzy numbers can be found in [1]. In the present paper we consider fuzzy numbers from a somewhat different perspective. Basically, we shall view fuzzy numbers in a topological vector space setting. Using the customary vector space operations together with the metric given in [4] we will define differentiation and integration of fuzzy-valued functions in ways that parallel closely the corresponding definitions for real differentiation and integration.

Topological properties of fuzzy numbers

Fuzzy numbers are defined and a metric is assigned to this class of fuzzy sets. Topological properties of the resulting metric space are studied. The notion of @q-crisp fuzzy numbers is introduced, and, in particular, it is shown that this class of fuzzy numbers is homeomorphic to the Hilbert cube.

Bases of fuzzy matroids

In this paper we continue our study of fuzzy matroids. We define the notion of a basis for a fuzzy matroid and investigate which properties of crisp matroid bases carry over to the fuzzy case.

Eigen fuzzy number sets

In this paper we extend certain results of Sanchez (Resolution of eigen fuzzy sets equations, Fuzzy Sets and Systems 1 (1978) 69-75) to the context of fuzzy numbers. A slight modification of the Dubois-Prade definition of fuzzy numbers is used.

Fuzzy rank functions

Abstract In this paper we obtain a characterization of fuzzy matroids in terms of the fuzzy rank function defined in [1]. This characterization parallels to some extent a well known characterization theorem for crisp matroids.

Fuzzy matroid sums and a greedy algorithm

Abstract In this paper we establish a necessary and sufficient condition for a greedy algorithm to find a maximal basis for a fuzzy matroid.

Introduction to fuzzy hypergraphs and Hebbian structures

Abstract Any finite collection of fuzzy subsets over a finite ground field determines a fuzzy hypergraph; subtract the word “fuzzy” and the statement defines a (crisp) hypergraph. From this inauspicious beginning, hypergraph theory has become an interesting and useful discipline; in this brief introduction, where serious development must occur elsewhere, a glimpse of what may be done within a fuzzy setting has been initiated. Of considerable importance is the question of applications. The focal point in this regard is an extended outline which expresses an implementation of fuzzy hypergraphs into Hebbian structures as introduced by Hebb (1949) in his distinguished treatise: The Organization of Behavior (A Neuropsychological Theory).

A note on the characterization of the max and min operators

Abstract In this note we obtain a characterization of the max and min operators, a characterization motivated by three simple set theoretic properties. The role of this characterization in the context of fuzzy set unions and intersections is also examined.

Fuzzy transversals of fuzzy hypergraphs

Abstract This is an introductory theory of fuzzy transversals associated with fuzzy hypergraphs. Adherence to details in order to provide familiarity with basic techniques and ideas has been a major goal of this paper.

Spanning properties for fuzzy matroids

Abstract In this paper we investigate the fuzzy analogs of some well known properties of the span of subsets of crisp matroids.