Fuzzy relational inequalities and equations, fuzzy quasi-orders, closures and openings of fuzzy sets

The paper deals with systems of fuzzy relation equations known in the literature as eigen fuzzy set equations, as well as with systems of related fuzzy relation inequalities. Our main results are algorithms for computing the greatest and the smallest solutions to the considered systems of fuzzy relation inequalities, and algorithms for computing the greatest solutions to the considered systems of fuzzy relation equations. The systems are studied in the framework of complete residuated lattices as the structure of membership values, by means of fuzzy quasi-orders and closures and openings on a collection of fuzzy sets.