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One application of isotopies of planar ternary rings

✍ Scribed by Dalibor Klucký

Publisher
Springer
Year
1975
Tongue
English
Weight
208 KB
Volume
3
Category
Article
ISSN
0046-5755

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✦ Synopsis

In this paper we are dealing with coordinatization of a given affine plane d by a PTR1). Such coordinatizations were first investigated in [4]. In [5] and [6] the necessary and sufficient conditions that a PTRZ coordinatizes a translation plane, a Desarguesian plane and a Pappian plane were deduced. These conditions were obtained by means of inner properties of the affine plane coordinatized by a given PTRZ. As it was pointed out independently by G. Pickert and V. Havel it offers also a routine method to deduce these conditions namely by use of isotopies between PTRZ's and HPTR's. In this article the method just mentioned will be realized.

This article is one of the results of research directed by Vficlav Havel.

1. SUMMARIZATION OF SOME WELL KNOWN RESULTS

Suppose that an affine plane d is coordinatized by a HPTR (S, t) and denote by "t and + t the induced operations defined by usual way2). It is known that: (I) d is a translation plane ¢,-

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