Lattice-valued approach to closed sets under fuzzy relations: Theory and applications

Motivated by fuzzy control problems and by some investigations of eigen fuzzy sets, we deal with a closedness of fuzzy sets under fuzzy relations in two ways: in one sense by directly analyzing fuzzy concepts and in the other by investigating the corresponding crisp problems in the cutworthy framework. Our main task is to investigate particular fuzzy functional equations and inequations appearing in this context, which turn out to be essentially connected with fuzzy control problems. We analyze procedures and find solutions of these equations and inequations, pointing to important applications.