| This paper discusses the existence of minimal solutions of fuzzy relationalequations with sup-conjunctor composition on complete Brouwerian lattices. First, for thefuzzy relational equations defned in a fnite domain, a formula that the number of minimalsolutions is given when every solution has a minimal solution. Then, a necessary and sufcientcondition that the fuzzy relational equations have minimal solutions is obtained when thesolution set is nonempty. Further, the solution set of fuzzy relational equations is found.Secondly, for the fuzzy relational equations assigned over an infnite domain, some propertiesof the solutions of fuzzy relational equations are discussed. Then, a sufcient condition that asolution has minimal elements in the solution set is shown if the solution set of fuzzy relationalequation is nonempty. Finally, some conditions that the intersection of two solutions of thefuzzy relational equations is also a solution are investigated on [0,1]. |
