System of fuzzy relation equations as a continuous model of IF–THEN rules

We claim that a fuzzy relation which models a system of fuzzy IF–THEN rules has the property of continuity only if the underlying system of fuzzy relation equations is solvable. In view of this connection, it is important to consider the problem of solvability of a system of fuzzy relation equations – with respect to an unknown fuzzy relation. This problem is analyzed in the present paper. We propose several new necessary and sufficient criteria for the so-called Mamdani relation to be a solution to the system. We prove that in the general case, these same criteria are sufficient (but not always necessary) for the solvability of the system. A new, easy-to-check criterion is proposed for the solvability of the system with special fuzzy parameters.