Wyniki wyszukiwania - BazTech

1

A residual skewaffine plane of a Mobius or Minkowski plane

Matraś A.

Demonstratio Mathematica

|

2007

|

Vol. 40, nr 4

913-924

EN

For Mobius and Minkowski planes of characteristic different from 2 a residual skewaffine piane associated with any point p is constructed. Following the construction given by Andre (cf. [1]) we obtain the residual piane as the group space of some normally transitive group of automorphisms fixing p. This is a skewaffine piane without straight lines in the Mobius case and with two families of straight lines in the Minkowski case.

2

Nearaffine planes related to pseudo-ordered fields

Jakóbowski J.

Bulletin of the Polish Academy of Sciences. Mathematics

|

2002

|

Vol. 50, no 3

345-360

EN

The constructions of affine planes and Minkowski planes related to pseudo-ordered fields are given in [3] and [2], respectively. We here give some analogous construction for nearaffine planes. Like before, we shall use some functions f, g and determine some conditions on f, g, necessary and suffcient to get the required plane. The Veblen postulate has a particular meaning in nearaffine planes, so it is also considered in the work. Some special cases like the field of the reals and finite fields of odd order are investigated, too. We give some examples of such nearaffine planes and consider their particular automorphisms. Every Minkowski plane related to pseudo-ordered field F determines a nearaffine plane connected with F [2, Proposition 1, p. 187]. But only weaker version of the reciprocal statement is true, i.e. a nearaffine plane related to a pseudo-ordered field determines a hyperbola structure (i.e. Minkowski plane without touching axiom).

3

Multicentral automorphisms in geometries of circles

Jakóbowski J., Matraś A.

Bulletin of the Polish Academy of Sciences. Mathematics

|

2001

|

Vol. 49, no 4

417-432

EN

We consider three types of geometries of circles (Moebius plane, Laguerre plane and Minkowski plane, cf. [4) with respect to so-called multicentral automorphisms. An automorphism [phi] of any geometry of circles is central if it has a fix point P and [phi] becomes a central collineation in the derived projective plane M(P). For any central automorphism [phi] we try to establish the whole set of points R such that [phi] becomes a central collineation in M(R.). Than [phi] is called multicentral if this set contains at least two points. Moreover, [phi] is proper if existing of a point [R is not equal to P], is not caused by the fact that [phi] is central in M(P). There is no proper multicentral automorphism in a Moebius plane. The most interesting proper multicentral automorphisms are involutorial mappings: double homotheties in Minkowski planes, and (sigma, tau)homologies in Laguerre planes. We give some examples.

4

A geometrical characterization of Minkowski planes of order 3 and 4

Jakóbowski J.

Demonstratio Mathematica

|

1999

|

Vol. 32, nr 2

441-456

EN

In [5] H. A. Wilbrink proved that a certain class of Minkowski planes induce nearaffine planes. Of course, this class contains all Minkowski planes over fields. But only Minkowski planes of order 3 and 4 induce nearaffine planes which are also Minkowski planes (moreover they are affine planes, too.