INTERACTIVE0-ALGEBRAS AND FUZZY OBJECTS OF TYPEN

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Abstract
The aim of this paper is to raise some questions–and partly, also to answer them –in connection with two important problem groups of fuzzy mathematics: n-fuzzy objects and the sigma-properties of different interactive fuzzy structures. These questions are suggested by the analyzation of natural languages, the common sense thinking – which are typical fields where the most adequate mathematical model is a fuzzy one-especially by complex adjectival structures and subjective “verifying” processes, respectively. They have, however, a real practical significance also in the field of engineering, as, e.g., in learning machine problems.In the first part we try to point to the practical importance of the concept of fuzzy objects of type n (or n-fuzzy objects), from the aspect of modeling natural languages. A useful way to define n-fuzzy algebras, i.e., generalizing ordinary fuzzy algebras for n-fuzzy objects, is also given, with introducing an isomorphism mapping from the fuzzy to the n-fuzzy object space. As an example, R-n-fuzzy algebra is defined. Because of the isomorphic property of the above mapping the later studies can be restricted to ordinary fuzzy objects. In the second part some very basic concepts in connection with the sigma-properties of fuzzy algebras are given and some simple theorems are proved. These are quite important from the aspect of fuzzy learning processes, as their probability theoretic interpretation leads to several convergence theorems – which are not dealt with here, however. In this part we raise the concept of the quantified of a fuzzy algebra, and by means of this concept a close relation between interactive fuzzy and Boolean algebras is proved –a very different relation from that between Zadeh's original, noninteractive system and Boolean algebra. Although any presentation of complete application examples is not at all intended in this paper, finally some aspects of the application of the above results, especially in learning control algorithms, are given, the statements backed up by the experience of a simulation experiment going on at present.

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