Eskişehir Teknik Üniversitesi Bilim ve Teknoloji Dergisi B - Teorik Bilimler 2667-419X Eskisehir Technical University 10.20290/estubtdb.1753779 Algebraic and Differential Geometry Cebirsel ve Diferansiyel Geometri FUZZY CHARACTERIZATION OF ALPHA AND BETA PLANES ON THE KLEIN QUADRIC IN PG(5,2) VIA MAXIMAL FLAGS FUZZY CHARACTERIZATION OF ALPHA AND BETA PLANES ON THE KLEIN QUADRIC IN PG(5,2) VIA MAXIMAL FLAGS https://orcid.org/0000-0003-1517-3409 Karakaya Münevvere Mine ESKİŞEHİR OSMANGAZİ ÜNİVERSİTESİ https://orcid.org/0000-0001-6379-0546 Akça Ziya ESKİŞEHİR OSMANGAZİ ÜNİVERSİTESİ 08 25 2025 13 2 122 130 07 29 2025 08 09 2025 Copyright © 2010, Eskişehir Teknik Üniversitesi Bilim ve Teknoloji Dergisi B - Teorik Bilimler 2010 Eskişehir Teknik Üniversitesi Bilim ve Teknoloji Dergisi B - Teorik Bilimler

In this study, we examine the fuzzy structures of ∝ (alpha) and β (beta) planes on the Klein quadric in the projective spacePG(5,2). Utilizing a maximal flag construction and its intersection with the hyperplane , we define a hierarchicalmembership function based on fuzzy set theory. Each point of PG(5,2) is assigned a degree of membership in [0,1] accordingto its level in the flag, satisfying ​. Through this framework, we analyze three alpha planes and three beta planes passingthrough the base point , classifying them by their fuzzy equivalence. It is shown that two alpha planes are fuzzy equivalent,while the beta planes are distinguished by the fuzzy degrees of the lines they share with the base plane. This approach bridgescombinatorial projective geometry and fuzzy logic, enriching the geometric understanding of the Klein correspondencethrough fuzzification.

In this study, we examine the fuzzy structures of α (alpha) and β (beta) planes on the Klein quadric in the projective space PG(5,2). Utilizing a maximal flag construction (q,U_1^', U_2^',U_3^',U_4^',PG(5,2)) and its intersection with the hyperplane H^5 (2), we define a hierarchical membership function λ^('' )based on fuzzy set theory. Each point of PG(5,2) is assigned a degree of membership in [0,1] according to its level in the flag, satisfying a_1≥a_2≥a_3≥a_4≥a_5≥a_6. Through this framework, we analyze three alpha planes and three beta planes passing through the base point q=(0,0,0,1,0,0), classifying them by their fuzzy equivalence. It is shown that two alpha planes are fuzzy equivalent, while the beta planes are distinguished by the fuzzy degrees of the lines they share with the base plane. This approach bridges combinatorial projective geometry and fuzzy logic, enriching the geometric understanding of the Klein correspondence through fuzzification.

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