Abou Zayda, Abdelsalem

El-atrash, Mohammed S.

Institutional Repository of the Islamic University of Gaza

Journal of The Islamic University of Gaza,(Series of Natural Studies &Engineering) Vol.13, No.2, P155-161,2005 ISSN 1726-6807, http://www.iugaza.edu/ara/research/  ON PROPERTIES OF GEOMETRY OF TYPE Dn,k  Mohammed Suliman El-Atrash    Professor of mathematics Science College, Mathematics Department Islamic University of Gaza, Gaza, Palestine E-mail: matrash@mail.iugaza.edu  Abdelsalam Abou zayda Assistant professor of mathematics Math. Department,  Al-Aqsa University, Gaza, Palestine  E-mail : zayda8@hotmail.com Abstract: In this paper we present a general case of the class of geometries Dn,2, Dn,3 and Dn,4 that is a point-line geometry of type Dn,k(F) where k≥2, n≥k+3 and F is a field.  We define the varieties of the geometry and we prove that the geometry is a parapolar with diameter k+1.  The properties related to the relations between the varieties will be investigated. 1-  Introduction  In [5] and [7] it has been proved that the class of geomtries Dn,2, Dn,3 and Dn,4 are weak parapolar geometries with diameters 3, 4 and 5 respectively.  We prove in this paper that the general case Dn,k(F) is a parapolar geometry with diameter k+1 for k≥2 and n≥k+3 and we get the same results with respect to the relations between symplecta themselves and the relations between points and symplecta. x⊥ means the set of all points in P collinear with x, including x itself. A subspace of a point-line geometry Γ=(P, L) is a subset X⊆P such that any line which has at least two of its incident points in X has all of its incident points in X.  〈X〉 means the intersection over all subspaces containing X, where X⊆P.  Lines incident with more than two points are called thick lines, those incident with exactly two points are called thin lines.  The singular rank of a space Γ is the maximal number n (possibly ∞) for which there exist a chain of distinct subspaces ∅≠X0⊂X1⊂...⊂Xn such that Xi is singular for each i, Xi≠Xj , i≠j, for example rank(∅)=-1, rank({p})=0 where p is a point and rank(L)=1 where L a line.  In a point-line geometry Γ=(P, L), a path of length n is a sequence of n+1 (x0,x1,..,xn) where, (xi,xi+1) are collinear, x0  is called the initial point and xn is called the end point.  A geodesic from a point x to a point y is a path of minimal possible length with initial point x and end point y.  We denote this length by dΓ (x, y). Mohammed El-Atrash, Abdelsalam Abou zayda   156 A geometry Γ is called connected if and only if for any two of its points there is a path connecting them.  A subset X of P is said to be convex if X contains all points of all geodesics connecting two points of X.  A polar space is a point-line geometry Γ=(P, L) satisfying the Buekenhout-Shult axiom [1] : For each point-line pair (p, l) with p not incident with l; p is collinear with one or all points of l, that is ⎜p⊥∩l⎪=1 or else p⊥⊃l.  Clearly this axiom is equivalent to saying that p⊥ is a geometric hyperplane of Γ for every point p∈P.  A point-line geometry Γ=(P, L) is called a projective plane if and only if it satisfies the following conditions [2]:  (i) Γ is a linear space; every two distinct points x, y in P lie exactly on one line,  (ii) every two lines intersect in one point,   (iii) there are four points no three of them are on a line.  A point-line geometry Γ=(P, L) is called a projective space if the following conditions are satisfied:   (i) every two points lie exactly on one line ,  (ii) if l1, l2  are two lines l1∩l2 ≠∅, then 〈l1, l2〉 is a projective plane.    ( 〈l1, l2〉 means the smallest subspace of Γ containing l1 and l2.)  A point-line geometry Γ=(P, L) is called a parapolar space if and only if it satisfies the following properties:  (i) Γ is a connected gamma space,  (ii) for every line l; l⊥ s not a singular subspace,  (iii) for every pair of non-collinear points x, y; x⊥∩y⊥ is either empty, a single point, or  a non-degenerate polar space of rank at least 2.  If x, y are distinct points in P, and if ⎢x⊥∩y⊥⎪=1, then (x, y) is called a special pair, and if x⊥∩y⊥ is a polar space, then (x, y) is called a polar pair (or a symplectic pair).  A parapolar space is called a strong parapolar space if it has no special pairs [4]. 2.  Construction of Dn,k (F)  Consider the polar space ∆=Ω+(2n, F) that comes from a vector space V of dimension 2n over a finite field F=GF(k) with a symmetric hyperbolic bilinear form B. SI is the set of all totally isotropic i-dimensional subspaces of V; 1≤ i≤ n-2.  The two classes M1, M2 consist of maximal totally isotropic n-dimensional subspaces.  Two n-subspaces fall in the same class if their intersection is of odd dimension. ON PROPERTIES OF…  157                   The geometry of type Dn,k(F) is the point-line geometry (P, L), whose set of points P is the collection of all k-dimensional subspaces of the vector space V, and whose lines are the pairs (A, B) where A is k-1-dimensional subspace of k+1-subspace B – that is, the set of (k-1, k+1)-subspace of V.  A point C is incident with a line (A, B) if and only if A⊂C⊂B as a subspaces of V.  To define the collinearity, let C1 and C2 be two point (the points are the T.I k-spaces), then C1 is collinear to C2 if and only if the intersection of C1∩C2 =k-1-space and 〈C1, C2〉 =k+1-space. The elements of the classes G1 and G2 are Grassmannian geometries of type An-1,k for more details about these kinds of geometries see [6] and [3]  There are two kinds of symlecta (1) The first kind is the convex polar spaces A3,2 that represent the (k-2, k+2) subspaces of V.  Then symplecton S of kind A3,2 is the set of TI k-dimensional spaces that contain the TI k-2-dimensional space and contained in the TI k+2-dimensional space.  (2) The second kind of symplecta is the convex polar spaces of type Dn-k+1,1 , that represent the collection of all TI k-1-subspaces of V.  Thus this kind of symplecta is defined as the collection of all TI k-subspaces of V that contain such TI k-1-subspaces. Notation.  Let the map Ψ:P→V defined above, i.e., Ψ(p) is the T.I. k-dimensional subspace corresponding to the point p.  We will use Ψ for the An-3,kG1 G2 Dn,k(F)L P Dn-k+2,2 Ak+2,kDn-k+3,,3 Dn-2,k-2 Dn-1,k-1 Sk-1 Sk Sk+1 Sn-2 M1 M2 Sk-3 S2 S1 Mohammed El-Atrash, Abdelsalam Abou zayda   158rest of the varieties of the geometry; for example Ψ(Dn-1,k-1) is the T.I. 1-dimensional subspace corresponding to a geometry of type Dn-1,k-1(F).  The inverse map Ψ-1 will be used for the inverse. 3-The main result.  The following theorem represents the first part of the main result of this paper that is: Dn,k (F) is weak parapolar geometry of diameter k+1. 3.1 Theorem.  Let Γ=(P, L) be the geometry of type Dn,k(F). Thus:  i- Γ is of diameter k+1,.  ii- Γ is parapolar geometry. Proof: i  We shall use the mathematical induction for k≥2.  If k=2, then we have geometry of type Dn,2(F) with diameter 3 [5].  Now for all the cases k-1, k-2, ….., 4, 3, 2 the geometries Dn,k-1, Dn,k-2,….., Dn,4, Dn,3, Dn,2 have the diameters k, k-1, ….., 5, 4, 3 respectively.  Now we shall prove that the diameter of Dn,k is k+1.  Suppose that p and q are two arbitrary points of Dn,k ,that represent TI k-spaces Ψ(p)=C1=〈x1,x2,…,xk〉 and Ψ(q)=C2=〈y1,y2,…,yk〉. If 〈x1,x2,…,xk〉∩〈y1,y2,…,yk〉.=1-space, 2-space,……or (k-2)-space, then the two points lie in the geometries Dn,k-1 , Dn,k-2,……and Dn,2 respectively.  Thus by the hypotheses the diameter are k, k-1,….,3 respetively.  If 〈x1,x2,…,xk〉∩〈y1,y2,…,yk〉.=(k-1)-space, without loss of generality we take z1=x2=y2, z2=x3=y3,…..,zk-1=xk=yk, then 〈x1,x2,…,xk〉∩〈y1,y2,…,yk〉= 〈z1,z2,…,zk-1〉.  Thus we have two cases: 1-  x1⊥∩C2=C2 2-  x1⊥∩C2= 〈z1,z2,…,zk-1〉. In case 1, 〈x1,y1,z1,z2,…,zk-1〉 forms a (k+1)-space and (〈z1,z2,…,zk-1〉,〈x1,y1,z1,z2,...,zk-1〉) is the line that contains the two points.  Thus d(p,q)=1. In case 2, B(x1,y1)≠0 and C2 is contained in a maximal TI n-space K, say K=〈y1,z1,z2,...,zk-1,zk,zk+1,…zn-1〉.  Then x1⊥∩K is a hyperplane of K, say H=〈z1,z2,...,zk-1,zk,zk+1,…zn-1〉.  We have two (k+1)-spaces 〈x1,z1,z2,...,zk-1,zk〉 and 〈y1,z1,z2,...,zk-1,zk〉, then 〈z1,z2,...,zk-1,zk〉 is a point that is collinear to each of C1 and C2.  Thus d(p,q)=2.  If 〈x1,x2,…,xk〉∩〈y1,y2,…,yk〉=0-space, then we the following cases: 1-  xi⊥∩C2=C2, i=1, 2, …,k. 2- xi⊥∩C2=〈y1,y2,…,yi-1,yi+1,…,yk〉  and xj⊥∩C2=C2 where i≠j and i, j=1, 2,…,k. 3- xi⊥∩C2=〈y1,y2,…,yi-1,yi+1,…,yk〉, , i=1, 2, …,k. ON PROPERTIES OF…  159In the first two cases we have always d(p,q)≤k because at least one of the vectors xi we have xi⊥∩C2=C2. In case 3, the least number of vertices that can make a geodesic between p and q is equal k.  Since C2 is contained in a maximal TI n-space, then without loss of generality we can choose the points r1,r2,…,rk such that Ψ(r1)=〈z,x2,x3,…,xk〉, Ψ(r2)=〈z,y1,x3,…,xk〉, Ψ(r3)= 〈z,y1,y2,x4,…,xk〉,……., Ψ(rk)= 〈z,y1,y2,…,yk-1〉.where z is a vector in the n-space in which C2 is contained.  Thus p∼r1, r1∼r2,…,rk∼q i.e., d(p, q)=k+1 and diameter of Dn,k is k+1.   ii-  To prove that the geometry is parapolar, we first prove that Dn,k is gamma space.  Assume that l is a line containing two points p and q such that ψ(p)= 〈x1,x2,…,xk〉 and ψ(p)= 〈y1,y2,…,yk〉.  Then by the definition of collinearity, the two points form a TI (k+1)-space 〈z1,z2,…,zk-1,x1,y1〉, where z1=x2=y2, z2=x3=y3,…, zk-1=xk=yk.  Let s be a point not in l such that s is collinear to p and q, we shall prove that s is collinear to every point incident to l.  Since s is collinear to p and q, then s does not contain the TI (k-1)-space 〈z1,z2,…,zk-1〉 and form (k+1)-spaces with p and q.  Now assume that r is arbitrary point in l such that r≠p≠q, then ψ(r), ψ(s) contain the same (k-1)-space 〈z1,z2,…,zk-1〉 and they are contained in (k+1)-space 〈z1,z2,…,zk-1,x1,y1〉.  Thus s is collinear to r. Now for the completion of the proof we need to show that for any line l ; l⊥ is non-singular.  For this purpose we take the same above line i.,e., l=(〈z1,z2,…,zk-1〉, 〈z1,z2,…,zk-1,x1,y1〉).  Thus the points r, s can be chosen to be non-collinear and each of them is collinear to p and q.  Since ψ(s) is contained in a maximal n-space, then we take 〈z1,z2,…,zk-2,x1,y1〉, 〈z1,z2,…,zk-1,u〉 to the points r, s where u is a vector in the n-space not in ψ(s).  Now 〈z1,z2,…,zk-2,x1,y1〉∩〈z1,z2,…,zk-1,u〉 does not contain a (k-1)-space, this means that r is not collinear to s but 〈z1,z2,…,zk-1,x1,y1〉 and 〈z1,z2,…,zk-1,y1,u〉 form (k+1)-spaces, then r, s are collinear to each of p and q.  Thus l⊥ is non-singular.    ♦ 3.2 Corollary.  Dn,k  is a non-degenerate weak geometry. Proof.  By the definition of collinearity and By Theorem 3.1, Dn,k is a non-degenerate.  Since the geometry of type Dn,2 is a weak sub-geometry of Dn,k [ 5 ], then Dn,k  is a weak.    ♦ 4- Properties of Dn,k It has been proved that for any pair of distinct sympleta (S1,S2) of the class of geometries Dn,2, Dn,3, Dn,4, rank(S1∩S2)=-1, 0, 2, Moreover for any pair Mohammed El-Atrash, Abdelsalam Abou zayda   160of non-incident point and symplecton (p, S), we have rank(p⊥∩S)=-1,1,2 see [5] and [7].  In this Paper we prove the same result is satisfied for the general case Dn,k. Remark.  Each geometry of the class G1 or G2 of Dn,k  is denoted by AT, where T is the TI n-space that corresponds to such geometry and AD denotes to the symlecton of type A3,2, where D is the TI (k+2)-space that corresponds to such symplecton. 4.1 Theorem.  Let Γ be a geometry of type Dn,k, then:  1-  Let AD1,AD2 be two distinct symplecta of type A3,2 with then rank(AD1∩AD2)=-1,0,2. 2-  If S1 and S2 are symplecta of type Dn-1,1, then rank(S1∩S2)=-1, 0. 3-  If S is a symplecton of type Dn-1,1 and AD is a symplecton of type A3,2, then rank(S∩AD)=-1, 2. 4-  If (p, S) is a non-incidence pair of point and symplecton, then rank(p⊥∩S)=-1, 1, 2. Proof.  1- AT1 and AT2 are located in the same class (G1), so AD1 and AD2 are located in the located in G1, then we have two cases: a-  AD1, AD2 are symplecta of the same geometry AT, then it has three cases:  a1-  If D1∩D2≤(k-1)-space, then rank(AD1∩ AD2)=-1.  a2-  if D1∩D2=k-space, then rank(AD1∩AD2)=0.  a3-  if D1∩D2=(k+1)-space, then rank(AD1∩AD2)=2. b- AD1, AD2 are symplecta of different geometries AT1 and AT2 respectively, then T1∩T2 is a space of odd dimension and we have three cases:  b1-  For dim(T1∩T2)≤k-1, dim(D1∩D2)≤k-1, so rank(AD1∩AD2)=-1.  b2- For dim(T1∩T2)=k, dim(D1∩D2)≤k. Then rank(AD1∩AD2)=-1, 0.  b3- For dim(T1∩T2)>k, then either dim(D1∩D2)≤k and rank(AD1∩AD2)=-1, 0 or dim(D1∩D2)=k+1 and rank(AD1∩AD2)=2.  Then rank(AD1∩AD2)=-1, 0, 2. Now the case in which AT1 and AT2 are located in different classes is similar to that cases above i.e., rank(AD1∩AD2)=-1, 0, 2. 2-  In this case ψ(S1) and ψ(S2) correspond to (k-1)-spaces.  We have two cases:  i-  If (ψ(S1)∩ψ(S2))<k-2, then we cannot find a k-space that contains 〈ψ(S1),ψ(S2)〉, i.e., S1∩S2 =ϕ.  Thus rank(S1∩S2)=-1.  ii-  If ψ(S1)∩ψ(S2)=k-2 and 〈ψ(S1),ψ(S2)〉=k-space, then rank(S1∩S2)=0. ON PROPERTIES OF…  1613-  If S is a symlecton of type Dn-1,1 and AD is a symplecton of type A3,2, then we have two cases:  i-  ψ(S)∩D≤(k-2)-space, then we there is no a k-space that contains ψ(S) and contained in D, i.e., rank(S∩AD)=-1.  ii-  ψ(S)⊆D, then the number of different k-spaces in D that contains ψ(S) form a space of rank 3 i.e., rank(S∩AD)=2.    ♦ 4-  a- For any pair (p, AD) of a point p and a symplecton AD of type A3,2, there are two cases:  ai-  If dim(ψ(p)∩D)<k-1, then there is no any k-space contained in D and meets ψ(p) in a (k-1)-space i.e., ψ(p)∩D=ϕ and rank(p⊥∩S)=-1.  aii- If dim(ψ(p)∩D)=k-1, then set of k-spaces in D that contain the (k-1)-space ψ(p)∩D form a projective plane i.e., rank(p⊥∩S)=2. b-  For any pair (p, S) of a point p and a symplecton S of type Dn-k+1,1, there are two cases:  bi-  If dim((ψ(p)∩ψ(S))<k-2, then we cannot find any k-space contains ψ(S) and meets ψ(p) in a (k-1)-space i.e., rank(p⊥∩S)=-1.  bii- If dim((ψ(p)∩ψ(S))=k-2, and 〈ψ(p),ψ(S)〉=(k+1)-space, then there are two k-spaces meet ψ(p) in a (k-1)-space and contain ψ(S).  Since the two k-space form a (k+1)-space, then p⊥∩S is a line i.e., rank(p⊥∩S)=1.      References 1. Buekenhout, F. & Shult, E. E. “On the foundation of polar geometry” Geometriae Dedicata 3  , 155-170. (1974), 2. Cameron, P. J. “Projective and polar spaces” Published by the School of Mathematical Sciences, QMW Univ. of London, U.K., (1990). 3. Cohen, A. M. & Cooperstein, B. N. “A characterization of some geometries of Lie type” Geometriae Dedicata 15 , 73-105. (1983), 4. Cooperstein, B. N. “A characterization of some Lie incidence structures” Geometriae Dedicata 6, 205-258. (1977) 5. El-Atrash, Mohammed and Zayda, Abdelsalam “Properties of point-line geometry of type Dn,2(F)” Al-Aqsa University Journal Vol. 5, No. 1, 35-44 , (2001) 6. Hanssens, G. “A technique in the classification theory of point-line geometries” Geometriae Dedicata 15 (1987), 85-101. 7. Zayda, Abdelsalam “Embedding and hyperplanes of point-line geometry of type Dn,k ,k=2,3,4” Ph.D. Thesis, Ain Shams University, Cairo, Egypt. (2002). 

ON PROPERTIES OF GEOMETRY OF TYPE Dn,k

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