Analytical Solution Methods for Fuzzy Relational Equations

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 modelling 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.