Method of solution to fuzzy relation equations in a complete Brouwerian lattice
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 circle dot 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 circle dot 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. (C) 2001 Elsevier Science B.V. All rights reserved.
fuzzy relation fuzzy relation equation brouwerian lattice solution set fuzzy elementary equation minimal solution