This work investigates the problem of solving a fuzzy relation equation  where  b are given elements of a complete Brouwerian lattice and Jis an infinite index set. This problem is an extension problem of the work of Wang et al. [12, 20, 22, 24]. The solution set comprises attainable and unattainable solutions. Attainable solutions are proved to be completely determined by the maximum solution and all of the minimal solutions. Therefore, finding all of the minimal solutions is critical to solving the complete solution. Hence, this work presents an efficient algorithm for determining all of the minimal solutions, in the sense that each produced solution is a minimum and no duplicate solutions exist.

infinite fuzzy relation equations, sup-min composition, Brouwerian lattice, minimal solutions, attainable solutions.