ONE CONSTRUCTION OF AN AFFINE PLANE OVER A CORPS

6200 | P a g e c o u n c i l f o r I n n o v a t i v e R e s e a r c h A p r i l 2 0 1 6 w w w . c i r w o r l d . c o m ONE CONSTRUCTION OF AN AFFINE PLANE OVER A CORPS Phd.Candidate.Orgest ZAKA, Prof.Dr.Kristaq FILIPI 1 Department of Mathematics, Faculty of Technical Science, University of Vlora “Ismail QEMALI”, Vlora, Albania. gertizaka@yahoo.com 2 Department of Mathematics, Faculty of Mathematical Engineering, Polytechnic University of Tirana, Tirana, Albania. f_kristaq@hotmail.com ABSTRACT


INTRODUCTION. GENERAL CONSIDERATIONS ON THE AFFINE PLANE AND THE CORPS
In this paper initially presented some definitions and statements on which the next material.
Let us have sets , , , where the two first are non-empty. Definition 1.1: The incidence structure called a ordering trio =(, , ) where ∩=Ø and  ⊆  × .
Elements of sets  we call points and will mark the capitalized alphabet, while those of the sets , we call blocks (or straight line) and will mark minuscule alphabet. As in any binary relation, the fact (P, ℓ ) ∈  for P ∈ and for ℓ ∈ , it will also mark P  ℓ and we will read, point P is incident with straight line ℓ or straight line ℓ there are incidents point P.
The straight line ℓ defined by points P and Q will mark the PQ. A2: For a point P ∈, and straight line ℓ ∈  such that (P, ℓ ) ∉ , there is one and only one straight line ∈ , passing the point P, and such that ℓ ∩ r = Ø. A3: In here are three non-incident points to a straight line.
The number n in Proposition 1.4, it called order of affine plane =(, , ), it is distinctly that the less order a finite affine plane, is n = 2. In a such affine plane it is with four points and six straight lines, shown in Fig.1. 2) The second action ⊚ It is associative ; 3) The second action ⊚ is distributive of the first operation of the first ⊕.
2) K ⋇ = K − 0 K it is a subset of the stable of K about multiplication;

3)
K ⋇ , ⊚ is a group.   Defined in this way is an incidence relations

TRANSFORMS OF A INCIDENCE STRUCTURES RELATING TO A CORPS IN A AFFINE PLANE
such that (P, ℓ),   P ℓ. So even here, when pionts P is incidents with straight line ℓ, we will say otherwise point P is located at straight line ℓ, or straight line ℓ passes by points P.
It is thus obtained, connected to the corps K a incidence structure =(, , ). Our intention is to study it.
According to (1), a straight line ℓ its having the equation Condition (2) met on three cases: 1) 0  a K and K , that allow the separation of the sets  the straight lines of its three subsets 0, 1, 2 as follows: Otherwise, subset is a sets of straight lines ℓ ∈ ℒ with equation subset is a sets of straight lines ℓ ∈ ℒ with equation Whereas subset is a sets of straight lines ℓ ∈ ℒ with equation Hence the  a straight line ℓ ∈ is completely determined by the element d ∈ K such that its equation is x = d,  a straight line ℓ ∈ is completely determined by the element f ∈ K such that its equation is y = f and  a straight line ℓ ∈ is completely determined by the elements k ≠ 0 K , g ∈ K such that its equation is y = x ⊚ k ⊕ g. Proof. Let P=(p1, p2) and Q=(q1, q2). Fact that P ≠ Q means (p1, p2) ≠ (q1, q2).
For two cases (7) and (7') notice that, when p1 = q1 and p2 ≠ q2, there exists a unique straight line  with equation = of the form (3'), so a line  0. Consider now the case 3) p1 ≠ q1 and p2 ≠ q2. From the fact P, Q   we have: The second equation can be written in the form 1 − 1 ⊚ = 2 − 2 ⊚ , that bearing ≠ 0 and ≠ 0 (9) Regarding to the coordinates of point P we distinguish these four cases: According to this result, equation (2) take the form ⊚ − 1 −1 ⊚ 2 ⊚ ⊕ ⊚ = 0 , where , according (9), ≠ 0 . So, by the properties of group we have:

 K b
So, by the properties of group we have: This bearing 2 0,  K q and the system (8) take the form In a similar way b) it is shown that equation (2) take the form After e few transformations equation (2) take the form d3 ) q1  0 K =q2. In this conditions (8) bearing After e few transformations equation (2) take the form .
The fact that = 1 , 2 ∉ ℓ, It brings to 1 ≠ . But the fact that ℓ ∩ = Ø, it means that there is no point ∈ , that ∈ ℓ and ∈ , otherwise is this true In other words there is no system solution (19') since ∈ , that brings In case a) 1  which solution point R = (d, −d ⊚ α ⊚ β −1 ) ∈ ℓ ∩ r 2 . Also straight line r 2 it does not meet the demand ℓ ∩ r 1 = Ø.
In this way we show that, whenℓ ∈ exist just a straight line r, whose equation is that satisfies the conditions of Theorem.
Conversely proved Theorem 2.2 is true for cases 2) ℓ ∈ dhe 3) ℓ ∈ . THEOREM 2.3. In the incidence structure =(, , ) connected to the corp K, there exists three points not in a straight line.
Proof. From Proposition 1.5, since the corp K is unitary ring, this contains 0 K and 1 K ∈ K, such that 0 K ≠ 1 K . It is obvious that the points P = (0 K , 0 K ), Q = (1 K , 0 K ) and R = (0 K , 1 K ) are different points pairwise distinct . Since P ≠ Q, and 0 K ≠ 1 K , by the case 2) of the proof of Theorem 2.1, results that the straight line PQ ∈ ℒ 1 , so it have equation of the form y = f. Since P ∈ PQ results that f = 0 K . So equation of PQ is y = 0 K . Easily notice that the point R ∉ PQ.
Three Theorems 2.1, 2.2, 2.3 shows that an incidence structure =(, , ) connected to the corp K, satisfy three axioms A1, A2, A3 of Definition 1.2 of an afine plane. As consequence we have THEOREM 2.4. An incidence structure =(, , ) connected to the corp K is an afine plane connected with that corp.