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Sets of solution-set-invariant coefficient matrices of simple fuzzy relation equations. (English) Zbl 0649.04003
Let x, b be two fuzzy sets and A, A’ be two fuzzy relations such that the finite fuzzy equations $$x\circ A=b$$ and $$x\circ A'=b$$ hold, where “$$\circ ''$$ denotes the max-min composition, A, A’ are assigned and x is unknown. If the membership values of b are ordered in strictly decreasing sense, then the above equations are called simple. Let X and X’, respectively, be the sets of the solutions of the above equations. Using results of Wang Peizhuang, S. Sessa, the reviewer, and W. Pedrycz [Busefal 18, 67-74 (1984; Zbl 0581.04001)], the authors give an iff condition in order to have $$X=X'$$. Further results on the number of the distinct elements of X are established.
Reviewer: A.Di Nola

##### MSC:
 03E99 Set theory 03B52 Fuzzy logic; logic of vagueness 94D05 Fuzzy sets and logic (in connection with information, communication, or circuits theory)
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##### References:
  Cheng-zhong, Lo, Reachable equation set of a fuzzy relation equation, J. math. anal. appl., 103, 524-532, (1984) · Zbl 0588.04005  Czogala, E.; Drewniak, J.; Pedrycz, W., Fuzzy relation equations on a finite set, Fuzzy sets and systems, 7, 89-101, (1982) · Zbl 0483.04001  di Nola, A.; Sessa, S., On measures of fuzziness of solutions of composite fuzzy relation equations, (), 277-281 · Zbl 0566.04003  Gottwald, S., On the existence of solutions of systems of fuzzy equations, Fuzzy sets and systems, 12, 301-302, (1984) · Zbl 0556.04002  Higashi, M.; Klir, G.J., Resolution of finite fuzzy relation equations, Fuzzy sets and systems, 13, 65-82, (1984) · Zbl 0553.04006  Lettieri, A.; Liguori, F., Characterization of some fuzzy relation equations provided with one solution on a finite set, Fuzzy sets and systems, 13, 83-94, (1984) · Zbl 0553.04004  Pappis, C.P.; Sugeno, M., Fuzzy relational equations and the inverse problem, Fuzzy sets and systems, 15, 79-90, (1985) · Zbl 0561.04003  Pei-zhuang, Wang; Yuan, Meng, Relation equations and relation inequalities, (), 20-31  Pei-zhuang, Wang; Sessa, S.; di Nola, A.; Pedrycz, W., How many lower solutions does a fuzzy relation equation have?, Busefal, 18, 67-74, (1984) · Zbl 0581.04001  Sanchez, E., Resolution of composite fuzzy relation equations, Inform. and control, 30, 38-48, (1976) · Zbl 0326.02048  Sanchez, E., Solutions in composite fuzzy relation equations: applications to medical diagnosis in Brouwerian logic, (), 221-234  Sanchez, E., Solution of fuzzy equations with extended operations, Fuzzy sets and systems, 12, 237-248, (1984) · Zbl 0556.04001
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