A note on fibered projective plane geometry

On the Menelaus and Ceva 6-Figures in the Fibered Projective Planes

“…Theorem 9 (see [7]). Suppose that one has a fibered projective plane FP with base plane P. Let , , and be three collinear points in P and let ( , ), ( , ), ( , ) be three -points of FP.…”

“…In [7], the following theorems give an answer to the question, when does a finite fibered projective plane exist given a particular triangular norm?…”

“…Let P be an arbitrary crisp projective plane and ∧ a triangular norm. Then there exists some nontrivial fibered projective plane FP with base plane P if and only if one of the following holds: In [7], the definition and theorems related to -harmonic conjugate in a fibered projective plane whose base plane satisfies Fano's axiom are given.…”

“…Definition 8 (see [7]). Suppose that one has a fibered projective plane FP with base projective plane P. Choose the four…”

“…They explored the second possibility, and it turns out that the geometric structures involved are much richer and defined fibered projective planes and looked at fibered versions of theorems of Desargues and Pappus [6]. The role of the triangular norm in the theory of fibered projective planes and fibered harmonic conjugates and a fibered version of Reidemeister's condition were given in [7]. The authors considered different triangular norms in the theory of fibered projective planes.…”