An efficient procedure for solving a fuzzy relational equation with max-Archimedean t-norm composition

In the literature, a necessary condition for minimal solutions of a fuzzy relational equation with max-product composition shows that each of its components is either zero or the corresponding component's value of the greatest solution. In this paper, we first extend this necessary condition to the situation with max-Archimedean triangular-norm (t-norm) composition. Based on this necessary condition, we then propose rules to reduce the problem size so that the complete set of minimal solutions can be computed efficiently. Furthermore, rather than work with the actual equations, we employ a simple matrix whose elements capture all of the properties of the equations in finding the minimal solutions. Numerical examples with specific cases of the max-Archimedean t-norm composition are provided to illustrate the procedure.