A Hyperbolic Characterization of Projective Klingenberg Planes Basri C ¸ elik Abstract—In this paper, the notion of Hyperbolic Klingenberg plane is introduced via a set of axioms like as Affine Klingenberg planes and Projective Klingenberg planes. Models of such planes are constructed by deleting a certain number m of equivalence classes of lines from a Projective Klingenberg plane. In the finite case, an upper bound for m is established and some combinatoric properties are investigated. Keywords—Hyperbolic planes, Klingenberg planes, Projective planes. I. I NTRODUCTION When one mentions the plane geometries, it reminds of affine planes, projective planes and hyperbolic planes. The property that differs these planes from others is given by the relation of being parallel on the set of lines. In affine planes, only one parallel line can be drawn to a line from a point not lying on this given line (Euclid’s famous 5th postulate, [8]). In projective planes all lines intersect, that is we cannot mention of parallel lines. In hyperbolic planes, exactly k parallel lines (k ≥ 2) to a given line from a point not lying on this given line. In literature, there is a lot of work on these planes. Geometrical structures which are more general then affine and projective planes are obtained by taking a class of points instead of a point; a class of lines instead of a line and by reorganising the incidence relation [1]. For affine planes, this generalisation can be found in [3], and for projective planes, in [2]. There is no such generalisation in literature for hyperbolic planes. Our main aim in this work is to give such a generalisation for hyperbolic planes and to present this generalisation as a system of axioms. In this paper incidence structures are defined as in [5] and blocks are called lines. For any point P ,(P ) denotes the set of lines incident with the point P ,[P ] the cardinality of (P ), and [P ,Q] the number of lines joining P and Q.(l), [l], and [l,d] are defined dually. A Projective Klingenberg plane (PK-plane) is an incidence structure K =(P , L, I ) together with an equivalence relation o on P and L (called neighbour relation, and the equivalence class of P (resp. l) is denoted by <P> (resp. <l>)) such that (PK1) P øQ = ⇒ [P, Q]=1, ∀P ,Q ∈P (PK2) lød = ⇒ [l, d]=1, ∀l, d ∈L (PK3) There exists a projective plane K ∗ (the canonical image of K) and an incidence structure epimorphism Basri C ¸elik is with the Uludag University, Department of Mathematics, Faculty of Science, Bursa-TURKEY, email: basri@uludag.edu.tr ϕ : K −→K ∗ such that P oQ ⇐⇒ ϕ(P )= ϕ(Q), ∀P, Q ∈P lod ⇐⇒ ϕ(l)= ϕ(d), ∀l, d ∈L Axiom (PK3) is equivalent to (PK3)’Putting <P> I <l> iff there are Q, d with QoP , dol and QI d, the equivalence classes with respect to this incidence an ordinary projective plane K ∗ . In the above definition ø means “non-neighbouring”, and PK-planes are denoted by K =(P , L, I ,o). A point P is said to be near a line l and this is denoted by P ol whenever P oQ for some QI l. For any element x of K, we denote the neighbour class of x by x-class. Detailed information about PK-planes can be found in [2], [4]. One can easily show the following lemma. Lemma 1.1: Let K =(P , L, I ,o) be a PK-plane. Then (i) P ol ⇐⇒ ∃h ∈L, such that hol, and P I h (ii) hod ⇐⇒ ∃H i I h, ∃D i I d  H i oD i , H 1 øH 2 , D 1 øD 2 , h, d ∈L,H i ,D i ∈P ,i =1, 2. (iii) “P 1 ∈P ,l 1 ,l 2 ∈L,P 1 I l 1 ,l 1 ol 2 ” = ⇒∃P 2  P 2 I l 2 ,P 1 oP 2 . When |P ∪ L| is finite, the geometric structure is called finite. Now, we state a theorem for finite regular PK-planes which can be found in [10]. The original proof of this theorem for Hjelmslev Planes is due to Kleinfeld [12]. Drake and Lenz [6] observed that this proof remains valid for PK-planes: Theorem 1.1: Let K =(P , L, I ,o)be a PK-plane. Then there are natural numbers t and r which are called the parameters of K, with (i) | <P> | = | <l> | = t 2 , ∀P ∈P ,l ∈L (ii) |(P )∩ <l> | = |(l)∩ <P> | = t, ∀P I l (iii) Let r be the order of projective plane K ∗ . If t =1, we have r ≤ t (then K is called proper and we have t =1 iff K is an ordinary projective plane). (iv) [P ]=[l]= t(r + 1), ∀P ∈P , ∀l ∈L (v) |P| = |L| = t 2 (r 2 + r + 1). II. HYPERBOLIC KLINGENBERG PLANES A projective or affine Klingenberg plane (PK-, AK-plane) is a generalization of ordinary projective plane where two points may also be multiply joined or not joined at all (see[3]). Now we can give a definition for Hyperbolic-Klingenberg plane (HK-plane) and it is given as a generalization of an ordinary projective plane, too. World Academy of Science, Engineering and Technology International Journal of Mathematical and Computational Sciences Vol:2, No:7, 2008 429 International Scholarly and Scientific Research & Innovation 2(7) 2008 scholar.waset.org/1307-6892/3890 International Science Index, Mathematical and Computational Sciences Vol:2, No:7, 2008 waset.org/Publication/3890