A model for fuzzy plane projective geometry (Q1425424)

The authors exhibit an example of a fuzzy projective plane in the sense of \textit{K. C. Gupta} and \textit{S. Ray} [Fuzzy Sets Syst. 54, No. 2, 191--206 (1993; Zbl 0789.51002)]. Two remarks are in order here. Remark 1. \textit{L. Kuijken} and the reviewer criticize in [Fuzzy Sets Syst. 138, 667--685 (2003)] the definition of Gupta and Ray in that it not really entails a fuzzy notion. Indeed, the order relation on the unit interval is not used in the definition, and hence one would rather call it a colored geometry. Moreover, this colored geometry is nothing else than a semi-affine plane in the sense of Dembowski. Hence there is no reason to keep the notion of fuzzy projective plane in the sense of Gupta and Ray in the literature. Remark 2. One trivial example of semi-affine plane is a projective plane in which every object gets one color. It is exactly this example that the authors ``discover''. Moreover, they restrict to the Pappian case by only considering vector spaces over commutative fields. They prove some other trivial things such as Desargues' theorem and the explicit form of the intersection of two 2-spaces in a 3-space.