Minkowski near— planes

Finite Laguerre Near-planes of Odd Order Admitting Desarguesian Derivations

“…Hence a proper Minkowski near-plane of order n ≥ 5, that is, circles of lengths n and n − 1 actually occur, extends to a Minkowski plane of order n. Clearly, such an extension is unique. Minkowski near-planes of order 3 and 4 were also described by Rinaldi in [10]. We summarize her results.…”

“…To be more precise, a Laguerre near-plane of order n ≥ 3 is an incidence structure of n 2 points, circles and generators satisfying the axioms (G), (C) and (J) from above. This definition extends the terminology for Minkowski planes and Möbius planes adopted in [10] and [13], respectively, see Section 4. Laguerre near-planes occur as special Laguerre semi-planes in [11] but have not been further investigated there.…”

“…This question was answered by Rinaldi in [10], although the introduction of the so-called Minkowski near-planes of order n was motivated differently. More precisely, a Minkowski near-plane of order n ≥ 3 is an incidence structure of points and circles satisfying the axioms (P), (G) and (J) from above but where axiom (C) is replaced by: (C') Each circle intersects each generator in at most one point and contains either n or n − 1 points.…”

“…In these cases, we have much more comprehensive results about their restrictions to derived affine planes, see [14] and [10]. To begin with, a finite Minkowski plane of order n − 1 where n ≥ 3 is an integer consists of a set P of points and a set C of circles such that the following four axioms are satisfied:…”