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1
ONE CONSTRUCTION OF AN AFFINE PLANE OVER
A CORPS
Phd.Candidate. Orgest ZAKA1, Prof.Dr.Kristaq FILIPI2
1University of Vlora “Ismail QEMALI”, Email Address gertizaka@yahoo.com
2 Polytechnic University of Tirana, Email Address f_kristaq@hotmail.com
Abstract.
In this paper, based on several meanings and statements discussed in the literature, we intend
constuction a affine plane about a of whatsoever corps (K,,). His points conceive as
ordered pairs (α,β), where α and β are elements of corps (K,,). Whereas straight-line in
corps, the conceptualize by equations of the type xayb=c, where a≠0K or b≠0K the
variables and coefficients are elements of that body. To achieve this construction we prove
some theorems which show that the incidence structure A=(P, L, I) connected to the corps
K satisfies axioms A1, A2, A3 definition of affine plane. In all proofs rely on the sense of the
corps as his ring and properties derived from that definition.
Keywords: The unitary ring, integral domain, zero division, corps, incdence structurse, point connected to a
corp, straight line connected to a corp, affine plane.
1. INTRODUCTION. GENERAL CONSIDERATIONS ON THEAFFINE PLANE
AND THE CORPS.
In this paper initially presented some definitions and statements on which the next material.
Let us have sets P,L,I, where the two first are non-empty.
Definition 1.1: The incidence structure called a ordering trio A=(P, L, I) where P∩L=Ø
and I P × L.
Elements of sets P we call points and will mark the capitalized alphabet, while those of the
sets L, we call blocks (or straight line) and will mark minuscule alphabet. As in any binary
relation, the fact (P,ℓ ) I for P P and for ℓL, it will also mark P I ℓ and we will
read, point P is incident with straight line ℓ or straight line ℓ there are incidents point P.
(See [3], [4], [5], [10], [11], [12], [13], [14], [15]).
Definition 1.2. ([3], [8], [16]) Affine plane called the incidence structure A=(P, L, I), that
satisfies the following axioms:
A1: For every two different points P and Q P, there is one and only one straight
line ℓL, passing of those points.
The straight line ℓ defined by points P and Q will mark the PQ.
A2: For a point P
P
, and straight line
ℓL
such that (P,
ℓ
)
I
, there is one and only one
straight line , passing the point P, and such that ℓ ∩
= Ø.
A3: In here are three non-incident points to a straight line.
2
A1 derived from the two lines different of many have a common point, in other words two
different straight lines of or do not have in common or have only one common point.
In affine plane A=(P, L, I), these statements are true.
Proposition 1.1. ([3], [5]) In affine plane A=(P, L, I), there are four points, all three of
which are not incident with a straight line (three points are called non-collinear).
Proposition 1.2. ([3], [6],]) In affine plane A=(P, L, I), exists four different straight line.
Proposition 1.2. ([3], [8]) In affine plane A=(P, L, I), every straight line is incident with at
least two different points.
Proposition 1.3. ([3], [9]) In affine plane A=(P, L, I), every point is incidents at least
three of straight line.
Proposition 1.4. ([3]) On a finite affine plane A=(P, L, I), every straight line contains the
same number of points and in every point the same number of straight line passes.
Furthermore, there is the natural number , n ≥2, such that:
1) In each of straight line , the number of incidents is points with him is n.
2) For every point , of affine plane A=(P, L, I), it has exactly n + 1 straight
line incident with him.
3) In a finite affine plane A=(P, L, I), there are exactly n2 points.
4) In a finite affine plane A=(P, L, I), there are exactly n2 +n straight line.
The number n in Proposition 1.4, it called order of affine plane A=(P, L, I), 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.
AB
C
D
1
A
2
A
3
A
4
A5
A
6
A
Fig.1
Definition 1.3. ([1]). The ring called structures ,,, that has the properties:
1) structure ,, is an abelian group;
2) The second action It is associative ;
3) The second action is distributive of the first operation of the first .
In a ring B,, also included the action deduction – accompanying each , from B,
sums
well
Proposition 1.5 ([1], [7]). In a unitary ring ,,, having more than one element, the
unitary element 1 is different from 0.
Definition 1.4 ([1], [2]). Corp called rings ,, that has the properties:
3
1) K is at least one element different from zero.
2) 0 it is a subset of the stable of about multiplication;
3) ,
is a group.
THEOREM 1.1. ([2]) If ,, is the corp, then:
1) it is the unitary element (is the unitary ring);
2) there is no zero divisor (is integral domain);
3) They have single solutions in K equations and , where
and are two elements what do you want of .
2. TRANSFORMS OF A INCIDENCE STRUCTURES RELATING TO A CORPS IN
A AFFINE PLANE.
Definition 2.1. Let it be ,, a corps. A ordered pairs (α, β) by coordinates α, β
K, called
point connected to the corp K.
Sets K2 of points associated with corps K mark P.
Definition 2.2. Let be a, b, c∈K.. Sets
ℓ = {(x, y) ∈K2 | xayb=c, a ≠ 0K or b ≠0K } (1)
called the straight line associated with corps K.
Equations xayb=c, called equations of the straight line ℓ. Sets of straight lines
connected to the body K mark L. It is evidently that
P
∩
L=
∅
.
Definition 2.3. Will say that the point P=(
α
,
β
)
∈
P is incident to straight line (1), if its
coordinates verify equation of ℓ,
This means that if it is true equation αaβb=c. This fact write down
P ℓ.
Defined in this way is an incidence relations
I P
×
L,
such that ∀(P, ℓ), PIL ⇔ 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 A=(P, L, I). Our intention
is to study it.
According to (1), a straight line ℓ its having the equation
, where 0
K
a
≠
or 0
K
b
≠
. (2)
Condition (2) met on three cases: 1) 0
K
a
≠
and 0
K
b
=
; 2) 0
K
a
=
and 0
K
b≠; 3) 0
K
a≠ and
0
K
b≠, that allow the separation of the sets L the straight lines of its three subsets L0, L1, L2
as follows: ℓ|,0and0;3
ℓ|,0and0;4
ℓ|,0and0.5
Otherwise, subset is a sets of straight lines ℓ with equation
4
, where 0, where ;3′
subset is a sets of straight lines ℓ with equation
, where 0, where ;4′
Whereas subset is a sets of straight lines ℓ with equation
,where0and0;5′
where 10,;
Hence the
• a straight line ℓ is completely determined by the element such that its
equation is ,
• a straight line ℓ is completely determined by the element such that its
equation is and
• a straight line ℓ is completely determined by the elements 0, such
that its equation is .
From the above it is clear that Π,, is a separation of the sets of straight lines .
THEOREM 2.1. For every two distinct points P, Q
∈
P , there exist only one straight
line A∈L that passes in those two points.
Proof. Let P=(p1, p2) and Q=(q1, q2). Fact that P ≠ Q means
(p1, p2) ≠ (q1, q2). (6)
Based on (6) we distinguish three cases:
1) p1 = q1 and p2 ≠ q2;
2) p1 ≠ q1 and p2 = q2;
3) p1 ≠ q1 and p2 ≠ q2;
Let’s be straight line A∈L , yet unknown, according to (2), having the equation
, where 0
K
a≠or 0
K
b≠.
Consider the case 1) p1 = q1 and p2 ≠ q2. From the fact P, Q
∈
A we have:
But and , so, from the fact that , is abelian group, by Definition 1.4,
we get
0
From above, according to Theorem 1.1, corps K is complete ring, so with no divisor 0,
results
0
0
From this
a) if 0, we get
5
0
00 0
00
.
According to this result, equation (2) takes the form 0, where 0, otherwise
0 (7)
(since, being 0, it is element of group , , so
00 0.
b) if 0, and is element of group , , exists
, that get the results:
00
under which, equation (2) in this case take the form
, where 0
1 (7')
Here it is used the right rules simplifying in the group , , with 0, because
and 0.
For two cases (7) and (7') notice that, when p1 = q
1 and p2 ≠ q
2, there exists a unique
straight line A with equation of the form (3'), so a line A
∈
L0.
Case 2) and is an analogous way and achieved in the conlusion and in this
case there exists a unique straight line A with equation of the form (4'), so a line
A∈L1.
Consider now the case 3) p1 ≠ q1 and p2 ≠ q2. From the fact P, Q
∈
A we have:
.8
The second equation can be written in the form
, that bearing 0 and 0 (9)
Regarding to the coordinates of point P we distinguish these four cases:
a) p1 = 0
K
=p2. This bearing 10
K
q
≠
and 20
K
q
≠
. In this conditions (8) take the form
0
.
According to this result, equation (2) take the form
0,
where , according (9), 0. So, by the properties of group we have:
0
0
, where
0 (10)
b) p1 = 0
K
≠p2. This bearing 10
K
q
≠
. In this conditions, system (8) take the form
.
−This result, give the equation (2) the form
,
where besides 0
K
c≠, by (9), the 0
K
b≠. So, by the properties of group we have:
6
1
, where
0 (11)
c) p1 ≠ 0
K
= p2. This bearing 20
K
q
≠
, and the system (8) take the form
.
In a similar way b) it is shown that equation (2) take the form
where 0 . (12)
d) p1 ≠ 0
K
and p2≠ 0
K
. We distinguish four subcases:
d1 ) q1 =0
K
= q2. From the system (8) we have
00 and
After e few transformations equation (2) take the form
, where
0 (13)
d2 ) q1 =0
K
≠ q2. From the system (8) we have
0
and
, where
.
After e few transformations equation (2) take the form
, ku
0 (14)
d3 ) q1 ≠0
K
=q2. In this conditions (8) bearing
0
and
,where
.
After e few transformations equation (2) take the form
,
where
0 (15)
d4 ) q1 ≠0
K
and q2 ≠0
K
. If c=0
K
system (8) have the form
0
.
7
After e few transformations results that the equation (2) have the form
,
where
0 (16)
If c0
K
≠, system (8) , by multiplying both sides of his equations with 1
c−, this is transform
as follows:
1
From this equation (2) take the form
,
where 0 (17)
As conclusion, from the four cases (14), (15), (16) and (17), we notice that, when p1 ≠ q1
and p2 ≠ q2, there exists an unique straight line A with equation of the form
(5'), so a line A∈L2.
THEOREM 2.2. For a point P ∈ P and a straight line A
∈
L such that P∉A exists only
one straight line r∈L passing the point P, and such that Ar
∩
=∅.
Proof. Let it be ,. We distinguish cases:
a) 0 and 0;
b) 0 and 0;
c) 0 and 0;
d) 0 and 0;
The straight line, still unknown , let us have equation
,00 (18)
For straight line ℓ, we distinguish these cases: 1) ℓ; 2) ℓ; 3) ℓ
Case 1) ℓ. In this case it has equation .
The fact that ,ℓ, It brings to . But the fact that ℓØ, it means that
there is no point , that ℓ and , otherwise is this true
,ℓ. (19)
In other words there is no system solution
. (19’)
since , that brings
, where00 (20)
In case a) 0 and 0, from (20) it turns out that 0,
Then equation (18) take the form
8
0, where 00
• If 00, equation (18) take the form
00
Determined so a straight line with equation 0, that passing point
0,0, for which the system (19’) no solution, after his appearance:
0
0
• If 00, equation (18) take the form
00
that defines a straight line . In this case system (19’) take the form
0
0,
which solution point ,0ℓ. This proved that straight line It does not
meet the demand ℓØ.
• If 00, equation (18) take the form
that defines a straight line . In this case system (19’) take the form
0
,
which solution point ,ℓ. Also straight line it does
not meet the demand ℓØ.
In this way we show that, whenℓ exist just a straight line , whose equation is
0
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 A=(P, L, I) 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 and 1,
such that 01. It is obvious that the points 0,0,1,0 and 0,1
are different points pairwise distinct . Since , and 01, by the case 2) of the
proof of Theorem 2.1, results that the straight line , so it have equation of the form
. Since results that 0. So equation of PQ is 0. Easily notice that
the point .
Three Theorems 2.1, 2.2, 2.3 shows that an incidence structure A=(P, L, I) 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 A=(P, L, I) connected to the corp K is an afine
plane connected with that corp.
9
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