Ali Jaballah

Uniqueness results in the representation of families of sets by fuzzy sets

The important concept of identifying a fuzzy subset of a set R with its corresponding chain of level sets is widely used in the literature. We investigate in this paper the problem of characterization of all fuzzy subsets of R that can be identified with a given arbitrary family C of subsets of R together with a given arbitrary subset S of [0,1]. Although our main focus is on the problem of uniqueness, we start by giving necessary and sufficient conditions for both existence and uniqueness of such fuzzy subsets. We continue by obtaining several algebraic and topological properties under which uniqueness is guaranteed. To further support our results, we present some examples which we hope will shed more light on the above mentioned problem of identification. We finish by raising a few natural questions that are left as open problems for further investigation.

Alternative characterizations for the representation of families of sets by fuzzy sets

We investigate the representation of a given family of sets by a fuzzy set in such a way that the level sets of the fuzzy set are precisely the given sets. This concept of representation is widely used in the literature for the classification and study of fuzzy structures on the same underlying set. Negoita and Ralescu obtained two theorems, both giving necessary and sufficient conditions, one called the representation theorem and the second called the generalized representation theorem. Our aim in this paper is to take a closer look at both theorems, hence establishing some equivalent forms and several constructive remarks, including some directions for future investigations.

From fuzzy sets to the decompositions of non-rigid sets

The concept of identifying a fuzzy subset of a set R with its collection of level sets and its collection of membership values plays an important role in many real life applications. Such identification is possible only if the fuzzy set can be recaptured. This implies that uniqueness of the fuzzy set plays an important role. It has been recently shown that in order to have uniqueness it is necessary and sufficient that the collection S of membership values of the fuzzy set be a rigid set. With this in mind, the authors went on to show that if S has the min-max-property then S is a rigid set and left the converse as an open problem. We start this paper by constructing a counterexample that shows that rigid sets do not have to have the min-max-property. We then take a closer look at non-rigid sets by decomposing them into isomorphism-invariant components and into S-connected components in the hope of getting a characterization of non-rigid sets, which will lead to a characterization of rigid sets and hence of uniqueness of the fuzzy set. We also investigate, in the case of nonuniqueness, the number of fuzzy sets corresponding to a collection of level sets and a collection of membership values.

Uniqueness in the generalized representation by fuzzy sets

The concept of representation of a given family of sets by a fuzzy set in such a way that the level sets of the fuzzy set are precisely the given sets is widely used in the literature for the classification and study of fuzzy algebraic structures on the same underlying set. The problem of existence of such fuzzy sets was investigated by various authors. As far as we know, the problem of uniqueness of such fuzzy sets was not studied before. We establish, among other things, necessary and sufficient conditions under which uniqueness is guaranteed in the cases of complete as well as partial representations of the level sets of the fuzzy set.

Existence and uniqueness of fuzzy ideals

Let R be a commutative ring. The idea of identifying a fuzzy ideal of R with its corresponding chain of level ideals and its range of values, although widely used in the literature for the study of fuzzy ideals, has never been investigated in a general setting before. We investigate in this paper the more general problem of characterization of all those fuzzy ideals that can be identified with a given arbitrary family C of subsets of R together with a given arbitrary subset S of [0,1]. Necessary and sufficient conditions for the existence and uniqueness of such fuzzy ideals are established. In particular, we obtain that uniqueness always holds true for Artinian rings. Moreover, examples are presented to further support our results and shed more light on the above-mentioned problem of identification. We finish by raising a natural question that is left as an open problem for further investigation.

Finite fuzzy topological spaces

Abstract We compute for the first time, the number of fuzzy topologies defined on a finite set and having a small number of open sets. Certain cases, where the number of open sets is large, are also considered. Several well known results are obtained as corollaries. The paper is ended by some questions for future investigations.

Chains in lattices of mappings and finite fuzzy topological spaces

Abstract In this paper, we establish several results concerning chains in Y X , the lattice of mappings from a finite set X into a finite totally ordered set Y. We compute the total number and the cardinalities of several collections of chains. As a byproduct we determine the total number of chained Y-fuzzy topologies defined on X. Several related and other well known results are obtained as corollaries. Also some natural questions are presented for further investigations.

Orthogonal Symmetries and Reflections in Banach Spaces

Let be a Banach space. We introduce a concept of orthogonal symmetry and reflection in . We then establish its relation with the concept of best approximation and investigate its implication on the shape of the unit ball of the Banach space by considering sections over subspaces. The results are then applied to the space of continuous functions on a compact set . We obtain some nontrivial symmetries of the unit ball of . We also show that, under natural symmetry conditions, every odd function is orthogonal to every even function in . We conclude with some suggestions for further investigations.