Some Results Of The Solution Set Of Infinite Sup-product Fuzzy Relational Equations On [0, 1] Lattice

In this paper, the infinite fuzzy relational equations defined on [0,1] are investigated. First, some properties of the solution set of the equationA o X = b (where"o"denotes sup - product composition, A = (aj)j∈J, J is aninfinite set ) are given when the domain is an infinite set on [0, 1]. From thecoe?cients of the equation A?X = b, it is shown that a suffcient and necessarycondition for existence of an attainable solution or an unattainable solution.Further, when the solution set is nonempty, the structure of the solution setof the equation is given. Next, the properties of the solution set of the fuzzyrelational equation A o X = B (where"o"denotes sup - product composition,A = (aij)I×J, B = bi, (i∈I), I = {1,2}, J is an infinite set ) are givenwhen the domain is an infinite set on [0, 1]. A concept of partially attainablesolution is introduced. From the coe?cients of the equation A o X = B,a suffcient and necessary condition that the solution set is nonempty isobtained, and a suffcient and necessary condition for the existence of anattainable solution, an unattainable solution or a partially attainable solutionare shown. In the end, under some condition, minimal solutions are formulated,and the number of minimal solutions is obtained. Further, when the so-lution set is nonempty, the structure of the solution set of the equations is given.