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A comment on fuzzy linear systems. (English) Zbl 1050.15003
Two recent papers of {\it M. Friedman}, {\it M. Ma} and {\it 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 {\it M. Friedman}, {\it M. Ma} and {\it 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 {\it 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.

15A06Linear equations (linear algebra)
15B48Positive matrices and their generalizations; cones of matrices
08A72Fuzzy algebraic structures
15B33Matrices over special rings (quaternions, finite fields, etc.)
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