On addition and multiplication of points in a certain class of projective Klingenberg planes

Abstract

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

A Note on Projective Klingenberg Planes over Rings of Plural Numbers

“…In particular addition and multiplication of points on a line is defined geometrically and interpreted algebraically, by using the coordinate ring. This generalizes a result of Celik and Erdogan [4] for the case of dual numbers (m=2).…”

“…We immediately start with giving the following proposition which is analogue of a result given in [4]. The calculations in the proof of the proposition are based on similar calculations used in the coordinatization procedure for general PK-planes due to Keppens [11,12].…”

“…Finally, we give the definition of addition and multiplication of points on the line OU of M(A) in the sense of [4].…”

Kuaterniyon 3-Uzay Üzerine Bazı Sonuçlar

“…For more detailed information about Q it can be seen to [3,4]. So, we can find the intersection points of any two lines and the lines joining any two points in the space P(J′′) where J′′=H(Q₄,Jγ ii.…”

“…The 2-space is isomorphic to the projective Klingenberg plane given in [3,4]. In a projective Klingenberg plane, it is well known that two non-connected (non-neighbour in [3,4]) lines meet at a unique point. However, this situation is different in 3-space as two lines with this propery meet at least at two points (see the results at pages 7 and 8).…”