Do We Need Minimal Solutions of Fuzzy Relational Equations in Advance?

Minimal solutions play a crucial role in description of all solutions to a fuzzy relational equation. The reason is that all solutions form a convex set with respect to (fuzzy) set inclusion; therefore, having all extremal solutions, we can represent the entire solution set as a union of intervals bounded from above by the greatest solution and from below by the minimal solutions. However, when computing the intervals, we obtain many duplicate solutions. The obvious question is as follows: Is there another way of representing the solution set, for instance, without the need of having all the minimal solutions in advance? We provide the positive answer to this question.