On Cross-Ratio in some Moufang-Klingenberg Planes Atilla Akpinar and Basri Celik Abstract—In this paper we are interested in Moufang-Klingenberg planes M(A) defined over a local alternative ring A of dual numbers. We show that a collineation of M(A) preserve cross-ratio. Also, we obtain some results about harmonic points. Keywords—Moufang-Klingenberg planes, local alternative ring, projective collineation, cross-ratio, harmonic points. I. I NTRODUCTION In the Euclidean plane, Desargues established the funde- mantal fact that cross-ratio (a concept originally introduced by Pappus of Alexandria c.300 B.C) is invariant under projection [4, p. 133]. For this reason, cross-ratio is one of the most important concepts of projective geometry. In this paper we deal with the class (which we will denote by M(A)) of Moufang-Klingenberg (MK) planes coordinatized by a local alternative ring A := A (ε)= A + Aε (an alternative field A, ε/ ∈ A and ε 2 =0) introduced by Blunck in [8]. We will show that a collineation of M(A) given in [2] preserves cross-ratio. Moreover, we will obtain some results related to harmonic points. For more information about some well-known properties of cross-ratio in the case of Moufang planes or MK-planes M(A), respectively, it can be seen the papers of [10], [5], [9] or [8], [1]. The paper is organized as follows: Section 2 includes some basic definitions and results from the literature. In Section 3 we will give a collineation of M(A) from [2] and we show that this collineation preserves cross-ratio. Finally, we obtain some results on harmonic points. II. PRELIMINARIES Let M =(P, L, ∈, ∼) consist of an incidence structure (P, L, ∈) (points, lines, incidence) and an equivalence relation ‘∼’ (neighbour relation) on P and on L, respectively. Then M is called a projective Klingenberg plane (PK-plane), if it satisfies the following axioms: (PK1) If P,Q are non-neighbour points, then there is a unique line PQ through P and Q. (PK2) If g,h are non-neighbour lines, then there is a unique point g ∩ h on both g and h. (PK3) There is a projective plane M ∗ =(P ∗ , L ∗ , ∈) and an incidence structure epimorphism Ψ: M → M ∗ , such that the conditions Ψ(P ) = Ψ(Q) ⇔ P ∼ Q, Ψ(g) = Ψ(h) ⇔ g ∼ h Atilla Akpinar and Basri Celik are with the Uludag University, De- partment of Mathematics, Faculty of Science, Bursa-TURKEY, emails: aakpinar@uludag.edu.tr, basri@uludag.edu.tr. hold for all P,Q ∈ P, g,h ∈ L. A point P ∈ P is called near a line g ∈ L iff there exists a line h ∼ g such that P ∈ h. Let h, k ∈ L, C ∈ P, C is not symmetric to h and k. Then the well-defined bijection σ := σ C (k,h):  h → k X → XC ∩ k mapping h to k is called a perspectivity from h to k with center C. A product of a finite number of perspectivities is called a projectivity. An incidence structure automorphism preserving and re- flecting the neighbour relation is called a collineation of M. A Moufang-Klingenberg plane (MK-plane) is a PK-plane M that generalizes a Moufang plane, and for which M ∗ is a Moufang plane (for the exact definition see [3]). An alternative ring (field) R is a not necessarily associative ring (field) that satisfies the alternative laws a (ab)= a 2 b, (ba) a = ba 2 , ∀a, b ∈ R. An alternative ring R with identity element 1 is called local if the set I of its non-unit elements is an ideal. We are now ready to give consecutively two important lemmas related to alternative rings. Lemma 2.1: The subring generated by any two elements of an alternative ring is associative (cf. [12, Theorem 3.1]). Lemma 2.2: The identities x (y (xz)) = (xyx) z ((yx) z) x = y (xzx) (xy)(zx)= x (yz) x which are known as Moufang identities are satisfied in every alternative ring (cf. [11, p. 160]). We summarize some basic concepts about the coordinatiza- tion of MK-planes from [3]. Let R be a local alternative ring. Then M(R)=(P, L, ∈ , ∼) is the incidence structure with neighbour relation defined World Academy of Science, Engineering and Technology International Journal of Mathematical and Computational Sciences Vol:4, No:8, 2010 1202 International Scholarly and Scientific Research & Innovation 4(8) 2010 scholar.waset.org/1307-6892/7820 International Science Index, Mathematical and Computational Sciences Vol:4, No:8, 2010 waset.org/Publication/7820