Extremal solutions to fuzzy relation equations and inequalities with three unknowns.

A large number of studies have investigated the systems of fuzzy relation equations and inequalities, which have a much wider field of application. In this paper, we study several types of systems of fuzzy relation equations and inequalities consisting of a given family of k-ary fuzzy relations, where natural number k ≥ 2, and three unknown fuzzy relations over complete residuated lattices and meet-continuous lattices. Their solutions are triples of fuzzy relations. For the systems of fuzzy relation inequalities, we give the greatest solutions contained in a given triple of fuzzy relations, the least solutions containing a given triple of fuzzy relations, or give maximal solutions contained in or containing a given triple of fuzzy relations, or belonging to a given interval of triples of fuzzy relations over complete residuated lattices and complete meet-continuous lattices. For the systems of fuzzy relation equations, we present a method of computing maximal solutions contained in a given triple of fuzzy relations and a method of computing minimal solutions containing a given triple of fuzzy relations. Furthermore, we provide some conditions under which there exist the greatest solutions contained in and the least solutions containing a given triple of fuzzy relations. [ABSTRACT FROM AUTHOR]