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In this study we investigate the structures constructed by deleting a subplane from a projective Klingenberg plane. If the superplane and the subplane are infinite, then it can be easily seen that the remaining structure satisfies the conditions of a hyperbolic Klingenberg plane. In this study we show that the remaining structure is the hyperbolic Klingenberg plane if the inequality r ≥ m 2 + m +1+ √ m 2 + m +2h olds when the superplane and the subplane are fi nite andt, r and t, m are their parameters, respectively. MSC: 51D20; 05B25

Let PK2(Q(e)) be the projective Klingenberg plane coordinated by the dual quaternion ring Q(e )= Q + Qe = {x + ye | x, y ∈ Q} where Q is any quaternion ring. In this paper, we determine the addition and multiplication of the points on the line (0, 1, 0) ofPK2(Q(e)) as the image of some collineations of the plane PK2(Q(e)). To do this, we give the collineations Sa and La. Later we show that the addition and multiplication of the nonneighbor points on the line (0, 1, 0) can be obtained as the images under that Sa and La. MSC: 51C05; 51J10; 12E15

Let ( O , E , U , V ) Open image in new window be the coordination quadruple of the projective Klingenberg plane (PK-plane) coordinated with dual quaternion ring Q ( e ) = Q + Q e = { x + y e ∣ x , y ∈ Q } Open image in new window, where Q is any quaternion ring over a field. In this paper, we define addition and multiplication of points on the line O U = [ 0 , 1 , 0 ] Open image in new window geometrically, also we give the algebraic correspondences of them. Finally, we carry over some well-known properties of ordinary addition and multiplication to our definition.