Method of solution to fuzzy relation equations in a complete Brouwerian lattice |
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Authors: | Xue-ping Wang |
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Affiliation: | a Department of Mathematics, Sichuan University, Chengdu 610064, Sichuan, People's Republic of China;b Department of Mathematics, Sichuan Normal University, Chengdu 610066, Sichuan, People's Republic of China |
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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 AX=b (where A=(a1,a2,…,an) and b are known, and X=(x1,x2,…,xn)T is unknown) when its solution set is nonempty, and b has an irredundant finite join-decomposition. Further, we give the method to solve AX=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. |
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Keywords: | Fuzzy relation Fuzzy relation equation Fuzzy elementary equation Brouwerian lattice Minimal solution Solution set |
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