The collineations which act as addition and multiplication on points in a certain class of projective Klingenberg planes

Abstract

Let [InlineEquation not available: see fulltext.] be the projective Klingenberg plane coordinated by the dual quaternion ring [InlineEquation not available: see fulltext.] where Q is any quaternion ring. In this paper, we determine the addition and multiplication of the points on the line [InlineEquation not available: see fulltext.] of [InlineEquation not available: see fulltext.] as the image of some collineations of the plane [InlineEquation not available: see fulltext.]. To do this, we give the collineations [InlineEquation not available: see fulltext.] and [InlineEquation not available: see fulltext.]. Later we show that the addition and multiplication of the nonneighbor points on the line [InlineEquation not available: see fulltext.] can be obtained as the images under that [InlineEquation not available: see fulltext.] and [InlineEquation not available: see fulltext.]. MSC: 51C05, 51J10, 12E15. © 2013 Celik and Dayioglu; licensee Springer.