In [9]
the author has studied polarities in finite 2-uniform projective Hjelmslev planes. The present paper deals with polarities in finite n-uniform projective Hjelmslev planes (n >~ 2).
0. Introduction
In this paper polarities in finite n-uniform projective Hjelmslev planes are studied. In the first section we recall the definition of such incidence structures together with some properties which will be used later. In Section 2 we define a polarity as an involutory one-to-one mapping from the pointset onto the line-set and from the line-set onto the point-set such that incidence and neighbor relation are preserved. A general formula for the number of absolute points is obtained. In Sections 3 and 4 polarities in the pappian n-uniform PH-planes PH(2, GF(q) [t]/t") are studied in detail. Section 3 deals with orthogonal polarities. We obtain that there are two types with qn-l(q + 1) and qn+[n/2]-i (q + 1) absolute points respectively. We also investigate the configurations formed by the absolute points in each case. Section 4 deals with unitary polarities in PH(2, GF(qZ)[t]/t").
Here we get that there are q3ln-~)(q3 + 1) absolute points.