FUZZY CHARACTERIZATION OF ALPHA AND BETA PLANES ON THE KLEIN QUADRIC IN PG(5,2) VIA MAXIMAL FLAGS

Yıl 2025, Cilt: 13 Sayı: 2, 122 - 130, 25.08.2025

Münevvere Mine Karakaya , Ziya Akça

https://doi.org/10.20290/estubtdb.1753779

Öz

In this study, we examine the fuzzy structures of ∝ (alpha) and β (beta) planes on the Klein quadric in the projective space
PG(5,2). Utilizing a maximal flag construction and its intersection with the hyperplane , we define a hierarchical
membership function 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 ​. Through this framework, we analyze three alpha planes and three beta planes passing
through 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 bridges
combinatorial projective geometry and fuzzy logic, enriching the geometric understanding of the Klein correspondence
through fuzzification.

Anahtar Kelimeler

Vector space , Projective space , Fuzzy set , Klein quadric

FUZZY CHARACTERIZATION OF ALPHA AND BETA PLANES ON THE KLEIN QUADRIC IN PG(5,2) VIA MAXIMAL FLAGS

Öz

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.

Anahtar Kelimeler

Vector space , Projective space , Fuzzy set , Klein quadric

Kaynakça

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