Andrzej Matraś

The Shortest Path Problem for the Distant Graph of the Projective Line Over the Ring of Integers

The distant graph

The Symmetry Axioms in Laguerre Planes

G=G({\mathbb {P}}(Z), \vartriangle )

Transitive groups of axial homologies of hyperbola structures and Minkowski planes

G=G(P(Z),△) of the projective line over the ring of integers is considered. The shortest path problem in this graph is solved by use of Klein’s geometric interpretation of Euclidean continued fractions. In case the minimal path is non-unique, all the possible splitting are described which allows us to give necessary and sufficient conditions for existence of a unique shortest path.

Properties of Auto- and Antiautomorphisms of Maximal Chain Structures and their Relations to i-Perspectivities

We discuss the free cyclic submodules over an associative ring R with unity. Special attention is paid to those which are generated by outliers. This paper describes all orbits of such submodules in the ring of lower triangular 3 × 3 matrices over a field F under the action of the general linear group. Besides rings with outliers generating free cyclic submodules, there are also rings with outliers generating only torsion cyclic submodules and without any outliers. We give examples of all cases.

Minkowski planes admitting automorphism groups of small type

We introduce two axioms in Laguerre geometry and prove that they provide a characterization of miquelian planes over fields of the characteristic different from 2. They allow to describe an involutory automorphism that sheds some new light on a Laguerre inversion as well as on a symmetry with respect to a pair of generators.

Automorphisms of Symmetric and Double Symmetric Chain Structures

In the paper we propose a modification of the classical construction of the (Minkowskian) incidence structures based on permutation groups. Dropping out explicit assumptions concerning rigidity and transitivity (and assuming an arbitrary finite ”dimension”) we obtain a wider class of structures. Their geometrical properties are studied; in particular, we establish their automorphism groups and discuss some problems related to axiomatic characterization.

A Construction of a Skewaffine Structure in Laguerre Geometry

In this paper a typification of the automorphism groups of hyperbola structures based on the notion of axial homologies (i.e. automorphisms fixing two generators of the same kind) is given. For the class of hyperbola structures over half-ordered fields (cf. [6, 12]) the types of the full automorphism groups are determined.