On the definition and some conjectures of fuzzy projective planes by Gupta and Ray, and a new definition of fuzzy building geometries
✍ Scribed by L. Kuijken; H. Van Maldeghem
- Publisher
- Elsevier Science
- Year
- 2003
- Tongue
- English
- Weight
- 283 KB
- Volume
- 138
- Category
- Article
- ISSN
- 0165-0114
No coin nor oath required. For personal study only.
✦ Synopsis
Gupta and Ray (Fuzzy Sets and Systems 54 (1993) 191) introduced the notion of fuzzy projective planes. We show that an old theorem of Dembowski classiÿes all ÿnite fuzzy projective planes, that the examples in Gupta and Ray (Fuzzy Sets and Systems 54 (1993) 191) arise from ordinary projective planes and have much easier descriptions that can easily be generalized, and that there exist fuzzy projective planes not arising from projective planes in that way. We also solve the two conjectures that they state, and we note that the fuzzy sets in the model of fuzzy projective planes given by Gupta and Ray do not use the order of the interval [0; 1], thus showing that their deÿnition is not optimal from fuzzy set theoretic point of view. Finally, we present a new, very general, deÿnition of fuzzy building, and, as a special case, this class contains fuzzy projective planes and spaces, as introduced in Kuijken (Proc. 1999 EUSFLAT-ESTYLF Joint Conf., Palma de Mallorca, Spain, 1999, pp. 429 -432). We initiate the study of fuzzy buildings and show that it gives rise to nontrivial combinatorial questions in the theory of buildings.