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1
Content available remote Havliček-Tietze configurations in various projective planes
Jakóbowski J., Kacperek D.
Demonstratio Mathematica
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2014
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Vol. 47, nr 4
979--988
EN
A. Lewandowski and H. Makowiecka proved in 1979 that existence of the Havliček–Tietze configuration (shortly H - T) in the desarguesian projective plane is equivalent to existence in the associated field, a root of polynomial x2 + x + 1, different from 1. We show that such a configuration exists in every projective plane over Galois field GF(p2) for p ≠ 3. As it has been demonstrated, in a projective plane over arbitrary field F, each hexagon contained in H - T, satisfies the Pappus–Pascal axiom, even if F is noncommutative. Moreover, such a hexagon either is pascalian or has exactly one pair of opposite sides intersecting at a point collinear with two points not belonging to these sides. In particular, all such hexagons are pascalian iff char F = 2. For the (noncommutative) field of quaternions, we have determined the set of all roots of the mentioned polynomial. Every H - T is the special Pappus configuration, in which three main diagonals of the hexagon are concurrent.
2
Content available remote Some examples of denumerable pseudo-ordered fields and their applications to geometry
Jakóbowski J.
Demonstratio Mathematica
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2013
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Vol. 46, nr 2
247--256
EN
The paper deals with the following problems concerning pseudo-ordered denumerable fields: (i) a connection between Fermat’s two squares theorem and the unique pseudo-order in a finite field; (ii) properties of a proper pseudo-order determined by any prime number in the field of rational numbers; (iii) existence of a proper pseudo-order in every subfield of the sequence used to obtain the field of constructible numbers; (iv) some brief of applications of the latter pseudo-orders to construct new algebraic and geometric structures. In particular, we extend the known construction of finite nearfields or quasifields given by e.g. W. A. Pierce or P. Dembowski – to infinite cases.
3
Content available remote Extending nearaffine planes to hyperbola structures
Cudna-Salmanowicz K., Jakóbowski J.
Bulletin of the Polish Academy of Sciences. Mathematics
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2007
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Vol. 55, no 1
71-80
EN
H. A. Wilbrink [Geom. Dedicata 12 (1982)] considered a class of Minkowski planes whose restrictions, called residual planes, are nearaffine planes. Our study goes in the opposite direction: what conditions on a nearaffine plane are necessary and sufficient to get an extension which is a hyperbola structure.
4
Content available remote Some generalization of nearaffine planes
Jakóbowski J.
Bulletin of the Polish Academy of Sciences. Mathematics
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2005
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Vol. 53, no 1
87-97
EN
There are three kinds of Benz planes: Mobius planes, Laguerre planes and Minkowski planes. A Minkowski plane satisfying an additional axiom is connected with some other structure called a nearaffine plane. We construct an analogous structure for a Laguerre plane. Moreover, our description is common for both cases.
5
Content available remote Central automorphisms of Veblenian nearaffine planes
Cudna-Lasecka K., Jakóbowski J.
Bulletin of the Polish Academy of Sciences. Mathematics
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2005
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Vol. 53, no 3
337-347
EN
The paper deals with nearaffine planes described by H. A. Wilbrink. We consider their central automorphisms, i.e. automorphisms satisfying the Veblen condition, which become central collineations in connected projective planes. Moreover, a concept of central pseudo-automorphism is considered, i.e. some bijections in a nearaffine plane are not automorphisms but they become central collineations in the related projective planes.
6
Content available remote Theorems on products of central collineations with distinct centers or axes applied to the Benz planes
Jakóbowski J.
Demonstratio Mathematica
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2004
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Vol. 37, nr 3
639--653
EN
There are three kinds of the Benz planes: Mobius planes, Laguerre planes and Minkowski planes [2, 3, 7]. In any Benz plane an automorphism φ is central if φ has a fixed point P and becomes a central collineation in the projective derived plane induced by P. Such central automorphisms have been considered by many authors (cf. [8,13, 11, 12, 10]), in particular the automorphism groups were classified. Usually product of two central collineations without common center or common axis is not central. But in some special cases it is central [4]. In this paper we apply theorems concerning such special cases - to the Benz planes.
7
Content available remote Nearaffine planes related to pseudo-ordered fields
Jakóbowski J.
Bulletin of the Polish Academy of Sciences. Mathematics
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2002
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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).
8
Content available remote Multicentral automorphisms in geometries of circles
Jakóbowski J., Matraś A.
Bulletin of the Polish Academy of Sciences. Mathematics
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2001
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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.
9
Content available remote A geometrical characterization of Minkowski planes of order 3 and 4
Jakóbowski J.
Demonstratio Mathematica
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1999
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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.
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