Method of solution to fuzzy relation equations in a complete Brouwerian lattice

Abstract

Up to now, how to solve a fuzzy relation equation in a complete Brouwerian lattice is still an open problem as Di Nola et al. point out. To this problem, the key problem is whether there exists a minimal element in the solution set when a fuzzy relation equation is solvable. In this paper, we first show that there is a minimal element in the solution set of a fuzzy relation equation A⊙X=b (where A=(a 1,a 2,…,a n) and b are known, and X=(x 1,x 2,…,x n) T is unknown) when its solution set is nonempty, and b has an irredundant finite join-decomposition. Further, we give the method to solve A⊙X=b in a complete Brouwerian lattice under the same conditions. Finally, a method to solve a more general fuzzy relation equation in a complete Brouwerian lattice when its solution set is nonempty is also given under similar conditions.