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We discuss the projective line \(\mathbb {P}(R)\) over a finite associative ring with unity. \(\mathbb {P}(R)\) is naturally endowed with the symmetric and anti-reflexive relation “distant”. We study the graph of this relation on \(\mathbb {P}(R)\) and classify up to isomorphism all distant graphs \(G(R, \Delta )\) for rings R up to order \(p^5\), p prime.

The distant graph \(G=G({\mathbb {P}}(Z), \vartriangle )\) 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...

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...

In a symmetric 2-structure \({\Sigma =(P,\mathfrak{G}_1,\mathfrak{G}_2,\mathfrak{K})}\) we fix a chain \({E \in \mathfrak{K}}\) and define on E two binary operations “+” and “·”. Then (E,+) is a K-loop and for \({E^* := E {\setminus}\{o\}}\), (E *,·) is a Bol loop. If \({\Sigma}\) is even point symmetric then (E,+ ,·) is a quasidomain and one has the set \({Aff(E,+,\cdot) := \{a...

We classify symmetric 2-structures \({(P, \mathfrak{G}_1, \mathfrak{G}_2, \mathfrak{K})}\), i.e. chain structures which correspond to sharply 2-transitive permutation sets (E, Σ) satisfying the condition: “ \({(*) \, \, \forall \sigma, \tau \in \Sigma : \sigma \circ \tau^{-1} \circ \sigma \in \Sigma}\) ”. To every chain \({K \in \mathfrak{K}}\) one can associate a reflection...

We show that every symmetric 2-structure \({(P,\mathfrak G_1,\mathfrak G_2,\mathfrak K)}\) of the class (III) [cf. Karzel H et al. (Result. Math., submitted)] is point symmetric, i.e. any two orthogonal chains \({A,B \in \mathfrak K}\) intersect in exactly one point and that any two points \({a,b \in P}\) have exactly one midpoint m : = a * b (with \({\widetilde m(a) = b}\) where...

In Karzel et al. (J. Geom. 99: 116–127, 2009) we introduced for a symmetric Minkowski plane \({ {\mathfrak M} := (P,\Lambda,{\mathfrak G}_1,{\mathfrak G}_2) }\) an order concept by the notion of an orthogonal valuation for the circles of Λ and showed that there is a one to one correspondence between the valuations and the halforderings of the accompanying commutative field. Here...