Search
Search Results
-
On the non-Paschian ordered planes
We give a non-Paschian plane based on the property of betweenness which cannot be derived from an ordering of the points of a line. In this model...
-
An axiomatic look at the Erdős-Trost problem
The Erdős-Trost problem can be formulated in the following way: “If the triangle XY Z is inscribed in the triangle ABC —with X , Y , and Z on the sides BC ...
-
Elliptic Reflection Spaces
If one considers the set P of all point reflections of an elliptic geometry and the group Γ generated by P then one obtains an example of a...
-
Area functions on affine planes
Oriented area functions are functions defined on the set of ordered triangles of an affine plane which are antisymmetric under odd permutations of...
-
Ordered metric geometry
Metric geometry in the sense of Hjelmslev and Bachmann studies metric planes of a very general kind without any assumption about order, continuity...
-
Ordered groups and ordered geometries
The article studies the relationship between ordered groups and ordered geometries. To this end we consider metric planes of a very general kind...
-
On the simplicity of ordered geometry
The Pasch axiom P is shown to be equivalent, given the linear order axioms, to the conjunction of Pasch’s Theorem PT with the Weak Crossbar Theorem WCBT...
-
The non-planarity of K 5 and K 3,3 as axioms for plane ordered geometry
The Pasch axiom is shown to be equivalent, given the linear order axioms, to the conjunction of its outer form with the statement that K ...
-
Geometric character of domain and the Harnack inequality of solution of a singular parabolic equation
The geometric character of domains of solutions of a singular parabolic equation with the Neumann boundary condition are discussed in this work. We...
-
The calculus of reflections and the order relation in hyperbolic geometry
It is well known that the calculus of reflections (developed by Hjelmslev, Bachmann et al.) enables the derivation of a large part of Euclidean and...
-
Acute triangulation of a triangle in a general setting revisited
Axiom A16 from Pambuccian (Can. Math. Bull. 53, 534–541,
2010 ) is shown to be superfluous as it depends on axioms A1–A15. This provides a... -
Reflection Spaces and Corresponding Kinematic Structures
For a reflection space ( P , Γ) [introduced in Karzel and Taherian (Results Math 59:213–218,
2011 )] we define the notion “Reducible Subspace”, consider... -
Another Splitting of the Pasch Axiom
The Pasch axiom is shown to be equivalent, given the linear order axioms, to the conjunction of its inner form with a statement involving five...
-
Elliptic Reflection Structures, K-Loop Derivations and Triangle-Inequality
This paper is a part of our general aim to study properties of elliptic and ordered elliptic geometries and then using some of these properties to...
-
Eine geometrische Kennzeichnung der reellen absoluten Ebenen
A new geometric characterization of the real absolute planes is presented, which is based upon few and simple axioms concerning properties of a...
-
Weakly ordered plane geometry
A plane geometry is introduced, which although ordered such that both every line is densely linearly ordered without first or last element and such...
-
Positive definitions of segment congruence in terms of segment inequality
It is shown that segment congruence can be defined by means of a positive existential definition in terms of segment inequality in several geometries...
-
Halforders and automorphisms of chain structures
The concepts of halfordered set, halfordered permutation set and halfordered chain structure will be presented and their interplay investigated. For...