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A comment on fuzzy linear systems. (English) Zbl 1050.15003

Two recent papers of M. Friedman, M. Ma and A. Kandel [Fuzzy Sets Syst. 96, No. 2, 201–209 (1998; Zbl 0929.15004); comment and reply ibid. 140, 559–561 (2003); ibid. 109, 55–58 (2000; Zbl 0945.15002)] deal with systems of linear equations with fuzzy right-hand side. In formulation of Theorem 3 (or Lemma 2 in the latter paper), instead of the quantifier ‘for every Y’, the phrase ‘for arbitrary Y’ was used. If we neglect the proof, then we can suppose that Y is arbitrarily fixed, and therefore, the theorem appears to be false.

This note indicates that Example 3 in the former paper contradicts Theorem 3 (for arbitrarily fixed Y). However, in M. Friedman, M. Ma and A. Kandel’s reply [loc. cit.], the part of the proof of the theorem is repeated with an explanation of the used quantifier.

Another discussion of the mentioned theorem was presented by R. Tyrala [Linear systems with fuzzy solutions, to appear in: Proc. Warsaw Internat. Seminar Soft Comp., Warsaw, Poland, November 2003], where we can find the next opinion that the formulation of the theorem is misleading.

MSC:
15A06Linear equations (linear algebra)
15A48Positive matrices and their generalizations (MSC2000)
08A72Fuzzy algebraic structures
15A33Matrices over special rings