Finite Hjelmslev Planes with New Integer Invariants

Projective Hjelmslev planes (PH-planes) are a generalization of projective planes in which each point-pair is joined by at least one line and, dually, each line-pair has a nontrivial intersection. Multiply joined points (and multiply intersecting lines) are called neighbor points (and neighbor lines). By hypothesis, the neighbor relations of a PH-plane A are equivalence relations which induce a canonical epimorphism from A to a projective plane A If A is finite, there exists [4] an integer t such that the inverse image of every point and every line of A contains precisely t elements. If the order of A is r, we say that A is a (t, r) PH-plane. We are concerned with the problem of determining the spectrum S of all admissible pairs (t, r). Since the finite projective planes are simply the (1, r) PH-planes, our concern is with a generalization of the classical existence question for projective planes. Prior to this announcement, the only pairs (t, r) known to belong to S satisfy the requirements: (1) t is a power of r, (2) r is a prime power. Conversely, all such pairs do belong to S, and all arise as the invariants of the Desarguesian-Pappian PH-planes investigated by Klingenberg [5]. A deep theorem of Artmann [1] allows one to assert that (t, r) is in S if (1) holds and if r is the order of a projective plane. Whether this is any improvement over the previous result is, however, still uncertain. Nonexistence results to date are also few in number. The celebrated Bruck-Ryser Theorem gives infinitely many values of r for which (1, r) $ S. Clearly (1, r) $S implies (t, r) fiS for any t. Kleinfield [4] has observed that (f, r) G S with t ^ 1 implies t > r. Most recently, Drake [2] has proved that (t, r) G S with I ^t ¥=r implies that t = 4 or 8 or that r < t + 1 y/"(2t 43). The current note is written to announce the following two existence results: