The collineations which act as addition and multiplication on points in a certain class of projective Klingenberg planes
© Celik and Dayioglu; licensee Springer 2013
Received: 11 January 2013
Accepted: 5 April 2013
Published: 19 April 2013
Let be the projective Klingenberg plane coordinated by the dual quaternion ring where Q is any quaternion ring. In this paper, we determine the addition and multiplication of the points on the line of as the image of some collineations of the plane . To do this, we give the collineations and . Later we show that the addition and multiplication of the nonneighbor points on the line can be obtained as the images under that and .
MSC:51C05, 51J10, 12E15.
1 Introduction and preliminaries
In the plane geometry, there are three important classes: affine planes, projective planes and hyperbolic planes. In recent years, studies on the generalization of these classes are becoming more popular. In this paper, we study on the projective Klingenberg planes, which are generalizations of the projective planes. Now we give some required concepts from [1–4] for understanding projective Klingenberg planes. A ring is defined as a set R together with two binary operations + and ⋅, which we call addition and multiplication, such that the following axioms are satisfied:
R1: is an Abelian group;
R2: Multiplication is associative;
R3: Distributive laws holds.
A ring R with identity element is called local if the set I of its non-units forms an ideal.
A Projective Plane is a system in which the elements of are called points and the elements of ℒ are called lines together with an incidence relation ∈ between the points and lines such that
P1: If P and Q are distinct points, then there is a unique line passing through P and Q (denoted by or PQ);
P2: If l and m are any lines, then there exist at least one point on both l and m;
P3: There exists four points such that no three of them are collinear.
In any projective plane, it is well known that there is a unique point on any distinct line pair and if l and m are distinct lines, the intersection point of these lines is denoted by or lm.
A Projective Klingenberg plane (PK-Plane) is a system where is an incidence structure and ∼ is an equivalence relation on (called neighboring) such that no point is neighbor to any line and the following axioms are satisfied:
PK1: If P and Q are non-neighbor points, then there is a unique line passing through P and Q;
PK2: If l and m are non-neighbor lines, then there is a unique point on both l and m;
PK3: There is a projective plane and an incidence structure epimorphism such that and .
A point is called near a line (which is denoted by ) iff there exists a line such that .
An incidence structure automorphism preserving and reflecting the neighbor relation is called a collineation of Π.
Let Π be a PK-plane with canonical image . Choose a basis whose image in form a quadrangle. Let , , , and . Let , . Then the points and the lines of Π get their coordinates as follows:
If , let where , ;
If , let where and ;
If , let where , and (clearly );
If , then where , ;
If , , then where , ;
If , then where , (then ).
Then , , , , , , , and a point has coordinates . We note that if and only if , for , dually for lines.
Let R be a local ring and the set of the non-units is denoted by I. Now we recall a theorem and corollary which are constructed in  for Moufang-Klingenberg planes.
The PK-Plane given in Theorem 1.1 is denoted by and is called the PK-Plane coordinatized with (the local ring) R.
Finally we give the definition of dual quaternions, some theorems and a corollary from , which we use in the next section.
where ε represents any element not in Q.
The elements of are called as dual quaternions. Obviously, the unity of is 1.
Theorem 1.3 The non-unit elements of are in the form bε, for and if , , then is a unit and .
Theorem 1.4 The set of non-units is an ideal of .
is a local ring (and it is called as the dual local ring on Q);
Theorem 1.6 Neighbor relation ∼ is an equivalence relation over and ℒ in .
For every ; .
2 Two collineations of
In this section, we will define two transformations for the points and lines of and also we will show that these transformations are collineations. Similar transformations can be found in .
Now, we can give the following theorem.
Theorem 2.1 The transformations and defined above are collineations of .
Proof It must be shown that and are bijective and preserves the incidence and the neighbor relations.
we conclude that and preserves the neighbor relation. □
3 Addition and multiplication of points and their correspondences with collineations
In this section, we recall some definitions,theorems and results about geometric addition and multiplication of points on OU in from  and also we will determine same relations between , and geometric definitions of addition and multiplication of points on OU where is a base of .
is defined as the intersection point of the lines LV and OU where , , ;
is defined as the intersection point of the lines VN and OU where , , , .
Now we give a theorem which interprets the relation between the geometric addition and multiplication of points and the collineations , which are given in last section.
- (2)If X is any non-neighbor point to U on the line OU then there exist a such that . In this case
Therefore, and for any point X on where . □
Dedicated to Professor Hari M Srivastava.
This work was supported by the Commission of Scientific Research Projects of Uludag University, Project number UAP(F)-2012/23.
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