The Solution Sets Of Fuzzy Relational Ecquations With Inf-α_R Composition

This paper deals the solution sets of fuzzy relational equations with inf-αR compositionover[0,1],where aαRb={ MR,a≤b,b,otherwise,and MR is the greatest elemen-t of R.In the case that the domain is finite,it first discusses the fuzzy relational equation∧ j∈J(rjαRxj)= a,and constructs all maximal solutions and describes the solution set of the equation.Then,it gives necessary and sufficient conditions that the fuzzy relational equation RαRX = A has solutions,and also obtains its set of all solutions.In the case that the domain is infinite,it first shows necessary and sufficient conditions that the fuzzy relation equation RαRX = a has solutions,and presents the conditions for the existence of maximal solutions of the equation,and then provides necessary and sufficient conditions that every solution of RαRX = a has a maximal one.