Solving fuzzy relation equations with max-continuous t-norm composition graphically

Fuzzy relation equations and their applications have been investigated extensively. The commonly discussed and used types usually belong to the type of max-continuous t-norms composition. This paper studies fuzzy relation equations of this type in a general meaning. The properties of the equations, especially the properties of minimal solutions and the characteristic matrix are discussed. They are closely related to the resolution of the equations. The reason why a solution could be related to a graph is revealed. It is found that all minimal solutions are determined by the characteristic matrix or the corresponding graph. And, they can be derived by the minimal coverings of the graph. Two algorithms for deriving all minimal solutions are given based on the properties of the equations and properties of the graph. Two examples are also given to demonstrate the two algorithms. Besides, the efficiency of the two algorithms and the value of the paper are discussed as well.