ELSEVIER Fuzzy Sets and Systems 75 (1995) 117--118

FUZZY sets and systems

Preface

Equations and relations on ordered structures: Mathematical aspects and applications

In this second issue we present several topics concerning the state-of-art of fuzzy relation equation theory. Like the traditional calculus has the key role in mathematical analysis, fuzzy rela- tion calculus has its central role in fuzzy set theory. And its natural consequence is the study of fuzzy relation equations whose importance, already mentioned in many papers and several mono- graphs, lies in the fact that human knowledge, rep- resented via suitable assignment of fuzzy relations, may be viewed as a system of such equations whose solutions give inference from that body of know- ledge.

Although this mechanism of inference can be modelled by using different types of equations, in accordance with the problem under study, other fuzzy relation-based models appear in the literature as that proposed here by D. Dubois and H. Prade for diagnosis problems. Their model is based on the possibility theory and the so-called twofold fuzzy relations and fuzzy sets, capable of handling uncer- tainty in a much more expressive way. C.P. Pappis defined in 1991 a similarity measure of fuzzy sets and he applied this concept in fuzzy control. In his joint paper with N.J. Karacapilidis, he combines the upper bound with the lower bounds of the solution set of a fuzzy relation equation determin- ing the correlated grades of similarity.

The paper of L.M. Kitainik deals with new techniques, called "a-cut mapping" and "canonical decomposition mapping", to reconstruct the mem- bership values of a fuzzy set from its a-cuts.

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Applications to fuzzy relation equation topics like reduction problems and eigen fuzzy sets of a fuzzy relation are also given.

K. Cechl/trovfi gives an iff condition in order to have a unique solution from a linear system of fuzzy equations and a test-algorithm is proposed. R.A. Cuninghame-Green revisits polynomials defined on the algebraic structure (~,max, +), already studied and applied in his previous 199l paper. It is shown that all the algebraic processes may be achieved by linear-time algo- rithms.

S. Gottwald characterizes triangular norms which define suitably a metric for fuzzy sets. This concept is then applied to fuzzy relation equa- tions which, although they do not have real solu- tions, can be seen as possessing "approximate solu- tions".

D. Bankovi6, following the traditional approach, gives quite a general formula for reproducing all the solutions of a given Boolean equation while S. Rudeanu proposes an algorithm for the solution of a quadratic Boolean equation. J. Drewniak in his paper studies particular solutions of a max-min fuzzy equation, i.e. matrices which are reflexive, irreftexive, symmetric, antisymmetric, complete and transitive giving iff conditions of existence. The last type of solutions is also investigated by K. Wagen- knecht in the wide context of max-t fuzzy equations and systems of inequalities.

Our conclusive paper deals with fuzzy relational structures exposing their knowledge representation

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118 Pre/ace / Fuzz)' Sets and Systems 75 (1995) 11~118

and addressing vital aspects of their development. The proposed multilevel structures significantly enhance representation and allow one to cope dir- ectly with the problem in which they are used. The examples in the area of decision-making sum-

marized here are only few among all the possible instances.

Antonio Di Nola, Witold Pedrycz and Salvatore Sessa