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

Allahviranloo, Tofigh

Fuzzy Sets Syst. 140, No. 3, 559 (2003).

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.

Reviewer: Józef Drewniak (Rzeszów)

MSC:

15A06	Linear equations (linear algebraic aspects)
15B48	Positive matrices and their generalizations; cones of matrices
08A72	Fuzzy algebraic structures
15B33	Matrices over special rings (quaternions, finite fields, etc.)

Keywords:

linear equation; fuzzy number; fuzzy solution

Citations:

Zbl 0929.15004; Zbl 0945.15002