A new look at solving a system of fuzzy relational equations

Abstract Solving a system of fuzzy relational equations exactly has been studied for many years. A usual understanding on the solvable conditions is that the input fuzzy sets must be normal and pairwise disjoint. Such understanding is reexamined in this paper. We show that the usual pairwise disjoint condition is too conservative and a system of max- t fuzzy relational equations can be solved exactly when the input fuzzy sets are semi-overlapped, a condition commonly found in most rule-based system applications. In addition, the boundary condition of solvability with respect to the compactness of the input fuzzy sets is derived. If it cannot be satisfied, we show that the system of equations could still be solved almost exactly by specifying the t -norm as drastic product. The results have been applied to study the capacity of fuzzy relations.