The structure of $n$-uniform translation Hjelmslev planes

Abstract

Affine or projective Hjelmslev planes are called 1-uniform (also strongly 1-uniform) if they are finite customary affine or projective planes. If n > 1, an n-uniform affine or projective Hjelmslev plane is a (finite) Hjelmslev plane U with the following property: for each point P of ?, the substructure n1 p of all neighbor points of P is an (n 1)-uniform affine Hjelmslev plane. Associated with each point P is a sequence of neighborhoods lp C 2p C ... C np = 2X. For i k, ('i)j and (i k -).k are isomorphic. Then if 2I(i) denotes (i?)i_ 1(1), q... , I(n) is a periodic sequence of ordinary translation planes (all of order r) whose period is divisible by k. It is proved that if T 1 . 9 Tk is an arbitrary sequence of translation planes with common order and if n > k, then there exists a strongly n-uniform translation Hjelmslev plane U of width k such that I(i) T. for i < k. The proof of this result depends heavily upon a characterization of the class of strongly n-uniform translation Hjelmslev planes which is given in this paper. This characterization is given in terms of the constructibility of the n-uniform planes from the (n 1)-uniform planes by means of group congruences. Introduction. Hjelmslev planes are generalizations of customary affine and projective planes in which distinct lines may intersect in more than a single point. Throughout this paper, we will refer to a Hjelmslev plane as an H-plane. One calls a finite H-plane 1-uniform if it is a customary affine or projective H-plane: Received by the editors May 14, 1971. AMS (MOS) subject classifications (1970). Primary 05B25, 50D35; Secondary 05B30.