Decomposition and resolution of min-implication fuzzy relation equations based on S-implications

The problem of solving min-implication fuzzy relation equations based on S-implications is studied in this paper. Decomposition of min-implication fuzzy relation equations is given in a finite case. The solution existence problem of min-implication fuzzy relation equations is discussed, and for NS-implications, a new solvability criterion based upon “solution matrices” introduced in this paper is studied. It is shown that for each solution r of a min-implication fuzzy relation equation, there exists a maximal solution r* of this equation, such that r* is greater than or equal to r, whenever the solution set of this equation is nonempty. The complete solution set of min-implication fuzzy relation equation can be determined by the maximal solution set of this equation, which is finite. An effective method to solve min-implication fuzzy relation equations based on S-implication is proposed.