A Hyperbolic Characterization of Projective Klingenberg Planes

Transcription

1 Intenational Jounal of Matheatics Sciences Volue Nube 1 A Hypebolic Chaacteization of Pojective Klingenbeg Planes Basi Çelik Abstact In this pape, the notion of Hypebolic Klingenbeg plane is intoduced via a set of axios like as Affine Klingenbeg planes and Pojective Klingenbeg planes. Models of such planes ae constucted by deleting a cetain nube of equivalence classes of lines fo a Pojective Klingenbeg plane. In the finite case, an uppe bound f is established and soe cobinatic popeties ae investigated. Keywds Hypebolic planes, Klingenbeg planes, Pojective planes. I. INTRODUCTION When one entions the plane geoeties, it einds of affine planes, pojective planes and hypebolic planes. The popety that diffes these planes fo othes is given by the elation of being paallel on the set of lines. In affine planes, only one paallel line can be dawn to a line fo a point not lying on this given line (Euclid s faous 5th postulate, [8]). In pojective planes all lines intesect, that is we cannot ention of paallel lines. In hypebolic planes, exactly k paallel lines (k ) to a given line fo a point not lying on this given line. In liteatue, thee is a lot of wk on these planes. Geoetical stuctues which ae e geneal then affine and pojective planes ae obtained by taking a class of points instead of a point; a class of lines instead of a line and by eganising the incidence elation [1]. F affine planes, this genealisation can be found in [3], and f pojective planes, in []. Thee is no such genealisation in liteatue f hypebolic planes. Ou ain ai in this wk is to give such a genealisation f hypebolic planes and to pesent this genealisation as a syste of axios. In this pape incidence stuctues ae defined as in [5] and blocks ae called lines. F any point P,(P) denotes the set of lines incident with the point P,[P] the cadinality of (P ), and [P,Q] the nube of lines joining P and Q. (l), [l], and [l,d] ae defined dually. A Pojective Klingenbeg plane (PK-plane) is an incidence stuctue K =(P, L, I) togethe with an equivalence elation o on P and L (called neighbou elation, and the equivalence class of P (esp. l) is denoted by <P >(esp. <l>)) such that (PK1) P øq = [P, Q] =1, P,Q P (PK) lød = [l, d] =1, l, d L (PK3) Thee exists a pojective plane K (the canonical iage of K) and an incidence stuctue epiphis Basi Çelik is with the Uludag Univesity, Depatent of Matheatics, Faculty of Science, Busa-TURKEY, eail: basi@uludag.edu.t ϕ : K K such that PoQ ϕ(p )=ϕ(q), P, Q P lod ϕ(l) =ϕ(d), l, d L Axio (PK3) is equivalent to (PK3) Putting <P >I <l>iff thee ae Q, d with QoP, dol and QId, the equivalence classes with espect to this incidence an dinay pojective plane K. In the above definition ø eans non-neighbouing, and PK-planes ae denoted by K =(P, L, I,o). A point P is said to be nea a line l and this is denoted by Pol wheneve PoQ f soe QIl. F any eleent x of K, we denote the neighbou class of x by x-class. Detailed infation about PK-planes can be found in [], [4]. One can easily show the following lea. Lea 1.1: Let K =(P, L, I,o) be a PK-plane. Then (i) Pol h L, such that hol, and P Ih (ii) hod H i Ih, D i Id H i od i, H 1 øh, D 1 ød,h,d L,H i,d i P,i=1,. (iii) P 1 P,l 1,l L,P 1 Il 1,l 1 ol = P P Il,P 1 op. When P L is finite, the geoetic stuctue is called finite. Now, we state a thee f finite egula PK-planes which can be found in [10]. The iginal poof of this thee f Hjelslev Planes is due to Kleinfeld [1]. Dake and Lenz [6] obseved that this poof eains valid f PK-planes: Thee 1.1: Let K = (P, L, I, o)be a PK-plane. Then thee ae natual nubes t and which ae called the paaetes of K, with (i) <P > = <l> = t, P P,l L (ii) (P ) <l> = (l) <P > = t, P Il (iii) Let be the de of pojective plane K.Ift 1, we have t (then K is called pope and we have t =1iff K is an dinay pojective plane). (iv) [P ]=[l] =t( +1), P P, l L (v) P = L = t ( + +1). II. HYPERBOLIC KLINGENBERG PLANES A pojective affine Klingenbeg plane (PK-, AK-plane) is a genealization of dinay pojective plane whee two points ay also be ultiply joined not joined at all (see[3]). Now we can give a definition f Hypebolic-Klingenbeg plane (HK-plane) and it is given as a genealization of an dinay pojective plane, too. 10

2 Intenational Jounal of Matheatics Sciences Volue Nube 1 A hypebolic plane is a geoetic stuctue such that (A1) Thee ae at least two points on each line. (A) Two distinct points lie on one and only one line. (A3) Thee exist at least fou points, no thee of which ae collinea. (A4) Though each point X not on a line l thee pass at least two lines not eeting (paallel to) l. (A5) If a subset S of the points contains all points on the lines though pais of distinct points of S, then the subset S contains all points of the geoetic stuctue.(see [7],[9],[11],[13]). A Hypebolic Klingenbeg plane (HK-plane) H is a syste (P, L, I,,o), whee (P, L, I) is an incidence stuctue and o is an equivalence elation on P L(called neighbouing) such that no eleent of P (point) is neighbou to any eleent of L (line), is an equivalence elation on L (called paallelis) and H satisfies the following axios f all P, Q Pand g, h L: (HK1) P øq = [P, Q] =1, P, Q P (HK) l L= P Il, QIl, PøQ (HK3) Thee exists at least fou paiwise non-neighbou points, no thee of which ae collinea. (HK4) F each point-line pai (P, l), P øl thee ae at least two non-neigh-bouing lines though P paallel to l. (HK5) Thee exists an hypebolic plane H =(P, L, I ) and an incidence stuctue epiphis such that ϕ : H H PoQ ϕ(p )=ϕ(q), P, Q P lod ϕ(l) =ϕ(d), l, d L and if [g, h] =0then ϕ(g) ϕ(h). Vaious odels f hypebolic planes such as Poincae odels (see [9]), Sandle s odels (see [13]) and the extension of Sandle s odels (see [11]) have been developed. It is well known that if a line is deleted fo a pojective plane then the eaining substuctue fs an affine plane. Gaves [7], Kaya-Özcan [11] and Sandle [13] have given exaples of hypebolic planes obtained by deletion fo pojective planes. Sandle had shown that if thee non-concuent lines ae deleted fo a pojective plane then the eaining stuctue fs a hypebolic plane in the sense of Gaves [7]. Kaya- Özcan [11] has extended Sandle s constuction and showed that if lines no thee of which ae concuent ae deleted fo a pojective plane then the eaining stuctue fs an hypebolic plane. If K is a PK-plane and l is a line, by deleting all lines neighbou to l and all points nea to l, the eaining stuctue fs an affine-klingenbeg plane. Now we will adopt the ethod of [11] to obtain an HK-plane fo a PK-plane. Let K =(P, L, I,o) be an infinite PK-plane and l i L, i =1,,..., denote paiwise non-neighbou lines such that no thee of the ae concuent, K =(P, L, I,o) be substuctue obtained fo K eoving all lines l i, togethe with the points which ae nea to l i, f i =1,,.., and 3 is any natual nube. In sybols P = {P P P øq, Qol i, i =1,,..., } L = L\{d L i, dol i, i =1,,..., } In this pape we accept in any substuctue K, paallelis is defined with l 1 l : : l 1 and l intesect on eoved points. Lea.1: If K is infinite, then f any substuctue K of K, (i) K = ϕ (K ) is a hypebolic plane. (ii) Thee ae at least two paiwise non-neighbou points on each line. (iii) Thee exists at least fou paiwise non-neighbou points, no thee of which ae collinea. (iv) If i such that dol i then d is a eoved line. Poof: The lines ϕ(l i )=li,i =1,,..., f a set of lines such that no thee of the ae concuent in the pojective plane K. Since the lines l i ae eoved fo K togethe with points nea l i, the lines ϕ(l i )=li ae eoved fo K togethe with thei points. We denote the eaining substuctue by K. It is easy to show that K is fs a hypebolic plane (This is shown in [11] f finite stuctue unde soe assuptations). Togethe with (i) and Lea 1.1, (ii), (iii) and (iv) ae obtained easily. Thee.1: If K is any infinite PK-plane then K is an HK-plane. Poof: (HK), (HK3) and (HK5) ae obtained fo Lea.1. (HK1) is obtained fo (PK1). Since at least thee paiwise non-neighbou points ae eoved fo each line in K, thee exist at least thee paiwise non-neighbou lines which ae though a point P and paallel to l if P øl. III. FINITE HK-PLANES Let K be a finite PK plane with paaetes t, in the sense of Thee 1.1. Then thee ae +1 paiwise non-neighbou points on each line of K. Let K be a substuctue such that + which is obtained by eoving the paiwise non-neighbou lines l i, i =1,,..., no thee of which ae concuent togethe with the points nea l i. Since no thee of the paiwise non-neighbou lines l i ae concuent, any 1 of the intesect the eaining eoved lines at paiwise nonneighbou points in K and theefe 1 +1 which gives the estiction +. The following lea, which gives the basic cobinatical popeties of K, can be shown by easy coputations: Lea 3.1: Following popeties ae valid in any stuctue K, +: (i) Two non-neighbou points of K ae on exactly one line. 11

3 Intenational Jounal of Matheatics Sciences Volue Nube 1 (ii) (iii) (iv) Though each point of K thee pass exactly t( +1) lines and thee pass exactly +1 paiwise non-neighbou lines of K. Thee ae exactly t ( + +1 ) lines and thee ae exactly + +1 paiwise non-neighbou lines in K. Thee ae exactly t + t ( 1)( ) points and thee ae exactly + 1 ( 1) ( ) paiwise non-neighbou points in K. Collay 3.: Any line class of K passing though s nonneighbou cne points in K, has exactly t ( +1+s ) points in K. Any line of K has exactly +1+s paiwise non-neighbou points. Now we state a lea which can be poved easily by using Lea 3.. Lea 3.3: If any line of K consideed as a line of K contains s cne points then the nube of deleted paiwise non-neighbou points fo this line is s and s if is even, s +1 if is odd. Collay 3.3: Let n denote the iniu nube of paiwise non-neighbou cne point on a line class of K, and k denote the nube of paiwise non-neighbou points on a line in K. Then +1+n k 1 ( ) A point of K is called a cne point if it is an intesection point of any two non-neighbou lines in the set of eoved lines. A point of K as a point class is called a pope cne point if it is an intesection class of any two non-neighbou lines in the set of eoved lines fo K. It is tivial that any cne point-class cesponds to a pope cne point unde ϕ. Lea 3.: Any line class ( any line) of K contains at ost 1 paiwise non-neighbou cne points in K, accding to is even odd, espectively. Poof: Let l be a line of K and assue that it contains s pope cne points in K. F each pope cne point, thee exist exactly two eoved non-neighbou lines of K on that point by the definition of pope cne point. Since no thee of the eoved non-neighbou lines ae concuent, thee exist exactly s distinct lines with this popety. Then s because the total nube of such lines is. Futhee, if is an odd nube clealy s iplies s 1. Then poof is copleted by consideing the popety that any cne point-class cesponds to pope cne point unde ϕ which is given just befe Lea 3.. The following collaies ae iediate: Collay 3.1: Any line class of K contains at ost t t 1 cne points in K, accding to is even odd, espectively. if is even and if is odd. +1+n k 1 ( 1) Poof: Let l be a line of K. l consideed as a line of K has at least n paiwise non-neighbou cne points and theefe it intesects at least n paiwise non-neighbou eoved line at paiwise non-neighbou points in K. But all of these points of intesection ae deleted and theefe l contains at least +1 ( n) paiwise non-neighbou points in K.Wehave +1 + n k since at ost n paiwise non-neighbou points can be deleted fo a line. On the othe hand if l, consideed as a line of K contains s paiwise non-neighbou cne points then the nube of paiwise non-neighbou points deleted fo l is s. But in the case whee is an even nube s by Lea 3.3. Hence the nube of paiwise non-neighbou points on l in K is less than +1, that is, k +1 =. Siilaly, if is an odd nube, then s +1 by Lea 3.3. Consequently k = 1. 1

4 Intenational Jounal of Matheatics Sciences Volue Nube 1 non-neighbou points and theefe it contains exactly Poposition 3.1: Let n be the iniu nube of paiwise non-neighbou cne point on a line of K.If 3 + n + 1 (1 4 +5) then K is a HK-plane. Poof: (HK1) is obvious since PK1. A line of K contains at least +1+n paiwise non-neighbou points by Collay 3.3. Theefe (HK) is satisfied iff i.e. which is clealy valid since +1+n n +1 n 1 (1 4 +5) n 1 (1 13) > n +1 by the hypothesis (since is the de of soe pojective plane; ). (HK3) is tivially satisfied, since thee exists two non-neighbou, non-intesecting lines in K each of which contains at least two non-neighbou points. F a poof of (HK4) let l be a line of K and P be a point P øl. The nube s in the Lea 3.3 is also the nube of nonneighbou lines passing though P and not eeting l. Hence thee exists at least +1 such lines accding to as is even odd, espectively. Theefe (HK4) is satisfied iff 3. Ifwetakeϕ = ϕ K then we ust only show that K is a hypebolic plane. In [9] it is shown that, if 3 + n + 1 (1 4 +5) then K is hypebolic plane. IV. SOME COMBINATORIC PROPERTIES OF K Let l be a line of K and P a point not on l. If l contains s non-neighbouing cne points then thee axists s non-neighbou lines paallel to l and passing though P. Then fo Lea 3.5 thee exist at ost s and at least +1 paiwise non-neighbou lines paallel to l and passing though P accding to is even odd, espectively. Theefe the nube of non-neighbou lines passing though P and paallel to l is independent of the choice of P but choice of l. The lines of K can be classified accding to the nube of non-neighbou points. Let C s denote the set of all lines of K such that each line of it contains exactly s nonneighbou cne points. Then, fo the Collay 3. C s contains exactly + s +1 t ( + s +1) points. Thus fo the Lea 3., thee exist 1 k +1 1 ( +1) k line classes in K which ae C k,c k+1,,c C k,c k+1,,ck 1 accding to is even odd, espectively. Lea 4.1: Nube of the non-neighbou lines of K paallel to a line l in the class C s is ( 1) s. Poof: Since l C s contains +1 ( s) nonneighbou points and except l, non-neighbou lines pass though each of these points it is obvious that the nube of non-neighbou lines paallel to l in K is ( 1) s. Lea 4.: Let P be any point of K and p s denote the nube of non-neighbou lines pass though P belong to C s, and q s denote the nube of all paiwise non-neighbou lines of C s. Then, p s = +1 q s = + +1 ( ) s p s = ( ) s q s = ( 1) [ ( s q s = 1+ whee t is espectively. )] ( ), 1 accding to is even odd, V. CONSEQUENCE AND SOME QUESTIONS In this pape, the definition of HK-plane is given and it is shown that the stuctues, obtained by deletion fo a PKplane of neighbou classes of paiwise distinct non-neighbou lines, no thee of the collinea, ae HK-planes, unde soe assuptations. Since any subplane of a PK-plane contains at least thee paiwise distinct non-collinea, non-neighbou lines, the stuctue which constucted by deleting any subplane fo the supeplane will be HK-plane, if it satisfies HK3. 13

5 Intenational Jounal of Matheatics Sciences Volue Nube 1 In the finite case, what is the elation between the paaetes of the subplane and of the supeplane and what is the uppe bund of paaetes of subplane? Is thee a way to distinguish the subplane deleted Desaguesian HK-plane fo all othe HK-planes? REFERENCES [1] Bacon, P.Y. An intoduction to Klingenbeg planes, Second Edition, Published by P.Y.Bacon, Flida, USA. [] Bake, C.A., Lane, N.D., Lie, J.W. A codinatization f Moufang- Klingenbeg planes, Sion Stevin, 65, No 1-A, (1991), pp. 3. [3] Bake, C.A., Lane,N.D., Lie,J.W. An Affine chaacteization of Moufang Pojective Klingenbeg planes, Res. in Maths., 17, (1990), pp [4] Blunck, A. Coss-atios in Moufang-Klingenbeg planes. Geo. Dedicata., 43, (199), pp [5] Debowski, P. Finite Geoeties, Spinge, Belin, Heidelbeg, New Yk, (1968). [6] Dake, D.A., Lenz, H. Finite Klingenbeg planes, Abh. Math. Se. Univ. Habug, 44, (1975), pp [7] Gaves, L.M. A finite Bolyai-Lobachevsky plane, Ae. Math. Monthly, 69, (196), pp [8] Heath, Si T.L. The thiteen books of the eleents, Second Edition,Vol.1 (Books I and II), Dove Publications, New Yk,1956. [9] Ivesen, B. Hypebolic Goeety, Cabidge Univ. Pess, (199). [10] Jungnickel, D. Regula Hjelslev planes, J. Cob. The. Se A, 6, (1979), pp [11] Kaya, R., Özcan, E. On the constuction of Bolyai-Lobachevsky planes fo pojective planes, Rendiconti Del Seinaio Mateatico Di Bescia, 7, (198), pp [1] Kleinfeld, E. Finite Hjelslev planes, Ill. J. Maths., 3, (1959), pp [13] Sandle, R. Finite hoogeneus Bolyai-Lobachevsky planes, Ae. Math. Monthly, 70, (1963), pp