The Solution Set Structure Of Fuzzy Relation Equations With Max-*Composition On [0,1]

In this paper,the solution sets of fuzzy relation equations with max-*compo-sition are investigated on[0,1].Firstly,a necessary and sufficient condition that the equation with single variable is solvable is shown.And then,we discuss the single equation with mul-tiple variables and obtain a necessary and sufficient condition that the equation is solvable.When the solution set is nonempty,the minimal solutions and the number of minimal solution-s are obtained.Finally,a necessary and sufficient condition that the equation set is solvable is shown.When the solution set is nonempty,an algorithm for computing minimal solutions and the solution sets of the equation set are obtained.Then,we discuss the problem that the fuzzy relational equations with max-*composition has a unique minimal solution(resp.a u-nique solution)on[0,1].First,we list of the properly of solutions with max-*composition,then show the character matrix and present some necessary and sufficient conditions for the ex-istence of a unique minimal solution(resp.a unique solution)for the fuzzy relational equations with max-*composition.At last,the solution set structure of interval-valued fuzzy relational equations with max-*composition is discussed.The concepts of tolerable solution set,unit-ed solution set and controllable solution set are introduced,respectively,and the properties of the solution set are shown.The necessary and sufficient conditions for the nonempty tolera-ble solution sets(resp.united solution sets,controllable solution sets)of interval-valued fuzzy relational equations are given.When the tolerable solution sets(resp.united solution sets,con-trollable solution sets)are nonempty,an algorithm for computing minimal tolerable solution sets(resp.minimal united solution sets,minimal controllable solution sets)and the structures of the three types of solution set are obtained.