Let (O, E, U, V) be the coordination quadruple of the projective Klingenberg plane (PK-plane) coordinated with dual quaternion ring Q(ε)where Q is any quaternion ring over a field. In this paper, we define addition and multiplication of points on the line OU = [0, 1, 0] 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. MSC: 51C05; 51N35; 14A22; 16L30
Let PK 2 (Q(ε)) be the projective Klingenberg plane coordinated by the dual quaternion ring Q(ε) = Q + Qε = {x + yε | 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] of PK 2 (Q(ε)) as the image of some collineations of the plane PK 2 (Q(ε)). To do this, we give the collineations S a and L a . 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 S a and L a . MSC: 51C05; 51J10; 12E15
Abstract. Let Π = (P ,L,I) be a finite projective plane of order n, and let Π = (P ,L ,I ) be a subplane of Π with order m which is not a Baer subplane (i.e., n ≥ m 2 +m). Consider the substructure, where I 0 stands for the restriction of I to P 0 ×L 0 . It is shown that every Π 0 is a hyperbolic plane, in the sense of Graves, if n ≥ m 2 +m+1+ m 2 + m + 2. Also we give some combinatorial properties of the line classes in Π 0 hyperbolic planes, and some relations between its points and lines.
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