## Analytical solution methods for fuzzy relational equations.(English)Zbl 0970.03044

Dubois, Didier (ed.) et al., Fundamentals of fuzzy sets. Foreword by Lotfi A. Zadeh. Dordrecht: Kluwer Academic Publishers. Handb. Fuzzy Sets Ser. 7, 291-340 (2000).
Summary: Fuzzy relational equations are without doubt the most important inverse problems arising from fuzzy set theory, and in particular from fuzzy relational calculus. Indeed, the calculus of fuzzy relations is a powerful one, with applications in fuzzy control and fuzzy systems modeling in general, approximate reasoning, relational databases, clustering, etc. In this chapter, fuzzy relational equations are approached from an order-theoretical point of view. It is shown how all inverse problems can be reduced to systems of polynomial lattice equations. The exposition is limited to the description of exact solutions, and analytical ways are presented for obtaining the complete solution set when working in a broad and interesting class of distributive lattices. Ample literature pointers to approximate solution methods and application areas are provided.
For the entire collection see [Zbl 0942.00007].

### MSC:

 03E72 Theory of fuzzy sets, etc. 03-02 Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations