Resolution Of Some Fuzzy Relational Equations On Complete Brouwerian Lattices

First, the infinite fuzzy relational equations with sup-inf com-position are discussed. Some properties of them are investigated, and some nec-essary and sufficient conditions for the existence of attainable solutions (resp. unattainable solutions and partially attainable ones) are given when their solu-tion sets are nonempty, and the set of attainable solutions is described. Secondly, the problem of solving the fuzzy relational equations with inf-LT composition on complete Brouwerian lattices are investigated under finite domains (resp. infinite domains). When the right-hand sides of fuzzy relational equations are meet-irreducible elements or have an irredundant finitely meet-decomposition, some necessary and sufficient conditions for an attainable solution (resp. an unattain-able solution) are formulated and some properties of the attainable solution (resp. the unattainable solution) are shown. Further, the solution sets are given when they are nonempty. Then the concepts of characteristic matrices and strong attainable variables are introduced. The relationships between the maximal so-lutions and the covering of characteristic matrices of fuzzy relational equations with inf-Zr composition are considered. It is shown that solving the system is closely related with the covering problem and proved that maximal solutions of system correspond to the irredundant coverings of characteristic matrices. The necessary and sufficient conditions that the system has a unique solution or a unique maximal solution are given. In the end, the problem of solving the inf-LT composite finite interval-valued fuzzy relational equations over complete Brouwe-rian lattices is investigated. The notions of tolerable solution set, united solution set and controllable solution set are introduced, respectively, and their properties are given. Some necessary and sufficient conditions that tolerable solution sets (resp. united solution sets、controllable solution sets) are nonempty are given and the structures of the three types of solution set are shown when the right-hand sides are meet-irreducible or have irredundant finitely meet-decomposable.