Ordered Line and Skew-Fields in the Desargues Affine Plane

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Based on the works of E.Artin [1] and J. Lipman [9] on ordered skew fields, we prove that the skew field that the constructed over an ordered line in a Desargues affine plane is a ordered skew field. To prove this, we must also give the definition of ordered Desargues affine plane, based on the definition given by E.Artin [1], but in this case the ordered line is given a suitable definition without the use of coordinates. Two main results are given in this paper, namely, every skew field constructed over a skew field over an ordered line in a Desargues affine plane is an ordered skew field 6. and every finite skew field constructed over a skew field over an ordered line in a Desargues affine plane in R2 is a finite ordered skew field 7.

2. Preliminaries Let P be a nonempty space, L a nonempty subset of P. The elements p of P are points and an element ℓ of L is a line. Collinear points on a line L are denoted by [A, E, B], where E is between A and B. Given distinct points A, B, there is a unique line ℓAB such that A, B lie on ℓAB and we write ℓAB = A + B [22, p. 52]. An affine space is a vector space with the origin removed [2, §4.1, p. 391]. The geometric structure (P, L) is an affine plane,asubspaceofanaffine space, provided

1o For each {P, Q}∈P, there is exactly one ℓ ∈ L such that {P, Q}∈ ℓ. 2o For each P ∈ P, ℓ ∈ L, P < ℓ, there is exactly one ℓ′ ∈ L such that P ∈ ℓ′ and ℓ ∩ ℓ′ = ∅ (Playfair Parallel Axiom [12]). Put another way, if P < ℓ, then there is a unique line ℓ′ on P missing ℓ [13]. 3o There is a 3-subset {P, Q, R}∈P, which is not a subset of any ℓ in the plane. Put another way, there exist three non-collinear points P [13]. Anaffine plane is a projective plane in which one line has been distinguished [6]. For simplicity, our affine geometry is on the Euclidean plane R2 and incident lines ℓ, ℓ′ are represented by ℓ ∩ ℓ′ , ∅ (intersection). A 0-plane is a point, a 1-plane a line containing a minimum of 2 collinear points and a 2-plane is an affine plane containing a minimum of 4 points, no 3 of which are collinear. Anaffine geometry is a geometry defined over vector spaces V, field F (vectors are points and subspaces of V) and subsets P (points), L (lines) and Π (planes) [11, §7.5]. Desargues’ Axiom, circa 1630 [7, §3.9, pp. 60-61] [15]. Let A, B, C, A′, B′, C′ ∈P AC A′C′ and let pairwise distinct lines ℓk, ℓl, ℓm, ℓ , ℓ ∈ L such that A A′ C C′ ℓk k ℓl k ℓm and ℓ k ℓ and ℓ k ℓ . ′ ′ ′ ′ A, B ∈ ℓAB, A′B′ ∈ ℓA B , and B, C ∈ ℓBC, B′C′ ∈ ℓB C . ′ ′ AB BC A , C, A , C , and ℓ , ℓl, ℓ , ℓl. ′ ′ Then ℓAC k ℓA C .

′ ′ Example 1. The parallel lines ℓAC, ℓA C ∈ L in Desargues’ Axiom are represented in Fig. 1. In other words, the base of △ABC is parallel with the base of △A′B′C′, provided the restrictions on the points and lines in Desargues’ Axiom are satisfied. A Desargues affine plane is an affine plane that satisfies Desargues’ Axiom. Theorem 1. Pappus, circa 320 B.C. [3, §1.4, p. 18]. If [ACE] ∈ ℓEA, [BFD] ∈ ℓBD and ℓBD, ℓCD, ℓEF meet ℓDE, ℓFA, ℓBC, then [NLM] are collinear on ℓNM. ORDEREDLINE&SKEWFIELDINDESARGUESAFFINEPLANE 3

ℓk ℓl ℓm B′ bc ℓA′B′ ℓB′C′ ℓA′C′ A′ bc bc C′

B bc ℓAB ℓBC ℓAC A bc bc C

′ ′ Figure 1. Desargues: ℓAC k ℓA C ℓEA A bc C bc E bc

N L M ℓNM bc bc bc (Pappian line)

bc B bc F bc D ℓBD

Figure 2. Pappian Line: [NLM] ∈ ℓNM

Example 2. The lines ℓBD, ℓCD, ℓEF, ℓDE, ℓFA, ℓBC in Pappus’ Axiom are represented in Fig. 2. In that case, the points of intersection [NLM] lie on the line ℓNM.

The aﬃne Pappus condition in Theorem 1 has an eﬀective formulation relative to [NLM] on line ℓNM given by N.D. Lane [8], i.e.,

Aﬃne Pappus Condition [8]. Let E, C, A, B, F, D be mutually distinct points as shown in Fig. 2 such that A, B, C lie on ℓEA and B, F, D lie on ℓBD and none of these points lie on ℓEA ∩ ℓBD, ℓBD ∩ ℓNM or ℓNM ∩ ℓEA. Then

ℓCB ∩ ℓEF lies on ℓNM ⇒ ℓAB ∩ ℓED lies on ℓNM. ℓAF ∩ ℓCD lies on ℓNM )

This leads to the following result. 4 ORGEST ZAKA AND JAMES F. PETERS

Theorem 2. Affine Pappus Condition [8]. Iftheaffine Pappus condition holds for all pairs of lines ℓ, ℓ′ such that ℓ ∦ ℓ′, then the affine Pappus condition holds for all pairs ℓ, ℓ′ with ℓ k ℓ′. Every Desarguesian affine plane is isomorphic to a coordinate plane over a field [16] and every finite field is commutative [10, §3, p. 351]. From this, we obtain Theorem 3. [Tecklenburg] [16]. Every finite Desarguesian affine plane is Pappian.

3. Ordered Lines and Ordered Desargues Affine Plane An invariant way to describe an ’order’ is by means of a ternary relation: the point B lies ”between” A and C. Hilbert has axiomatized this ternary relation [5]. In this section, we begin by giving a suitable deﬁnition for our search for lines in Desargues aﬃne plane, based on the meaning of betweenness given by D.Hilbert, i.e., if B lies ”between” A and C, we mark it with [A, B, C].

Definition 1. An ordered line in a Desargues Affine plane (briefly, called the line) satisfies the following axioms. Lo.1 For A, B, C ∈ ℓ, [A, B, C] =⇒ [C, B, A] . Lo.2 For A, B, C ∈ ℓ are mutually distinct, then we have exactly one, [A, B, C] , [B, C, A] or [C, A, B] . Lo.3 For A, B, C, D ∈ ℓ , then [A, B, C] and [B, C, D] =⇒ [A, B, D] and [A, C, D] . Lo.4 For A, B, C, D ∈ ℓ , then [A, B, C] and [C, B, D] =⇒ [D, A, B] or [A, D, B] .

D (a)

bc bc bc bc bc B A C

D (b) bc bc bc bc bc B A C

B (c)

bc bc bc bc bc C A D

Figure 3. Possibilities of doubles combinations, of [B, A, C] , [C, A, D] and [D, A, B]

Proposition 1. For all A, B, C, D ∈ ℓ , two, of [B, A, C] , [C, A, D] and [D, A, B] exclude the third. Proof. Let’s get double combinations and see how third is excluded. ORDEREDLINE&SKEWFIELDINDESARGUESAFFINEPLANE 5

(a) Suppose we are true [B, A, C]and[C, A, D] , (see Fig. 3, (a)) then from the order axioms in Def. 1, we have:

[B, A, C] Lo.1 [B, A, C] Lo.4 =⇒ =⇒ [D, B, A] ∨ [B, D, A] . [C, A, D] ) [D, A, C] ) From Axiom Lo.2 loses the possibility that [D, A, B] is true, so [D, A, B] is false. (b) Suppose we are true [B, A, C] and [D, A, B] , (see Fig. 3, (b)) then from the axioms above we have:

[B, A, C] Lo.1 [B, A, C] Lo.4 =⇒ =⇒ [A, D, C] ∨ [A, C, D] . [D, A, B] ) [B, A, D] ) From Axiom Lo.2 loses the possibility that [C, A, D] be true, so [C, A, D] is false. but each of them [A, D, C] ∨ [A, C, D] derives that [C, A, D] is false. (c) Suppose we are true [B, A, C]and[D, A, B] , (see Fig. 3, (c)) then from the axioms above we have:

[C, A, D] Lo.4 =⇒ [B, C, A] ∨ [C, A, B] . [D, A, B] ) From Axiom Lo.2 loses the possibility that [B, A, C] be true, so [B, A, C] is false.

Definition 2. For three different points A, B, C ∈ ℓ , that, we say that points B and C lie on the same side of point A, if we have exactly one of [A, B, C] , [A, C, B] . Proposition 2. For every four different points A, B, C, D in a line ℓ, in Desargues affine plane. Then [A, B, C] [A, B, D] =⇒ [A, C, D] ) ( [B, C, D] Proof. By [A, B, C], we have that the points A and B are in the same side of point C, by [A, C, D], we have that the points A and D are on the opposite side of point C. Then, we have that, the point B and D are on the opposite side of point C (since otherwise we would have that points A, B and D would be on the same side of point C, this would exclude [A, C, D]), so we have that [B, C, D]. By [A, B, C], we have that the points A and C on the opposite side of point B. Since [B, C, D], we have that the points C and D are in the same side of point B. Then, we have that, the point A and D are on opposite sides of the point B (since otherwise we would have that points A and D would be on the same side of point B, and point C this will give us [D, B, C] this would exclude [B, C, D]), so we have that [A, B, D]. Definition 3. For three different points A, B, C ∈ ℓ , that, we say that points B and C lie on the same side of point A, if we have exactly one of [A, B, C] , [A, C, B] . Proposition 3. If we have, that, for four different points A, B, C, D ∈ ℓ: points B and C lie on the same side of point A, and the points C and D lie on the same side of point A, then the points B and D lie on the same side of point A. 6 ORGEST ZAKA AND JAMES F. PETERS

Proof. Let’s look at the possible cases of ’positioning’ between the points, If [A, D, C] Lo.1 [A, D, C] 2 =⇒ =⇒ [A, D, B] . [B, C, A] ) [A, C, B] ) If

[A, B, C] 2 =⇒ [A, B, D] . [A, C, D] ) If [A, D, C] 1 =⇒ [A, B, D] ∨ [A, D, B] . [A, B, C] ) If [A, D, C] Lo.1 [A, D, C] Lo.3 =⇒ =⇒ [A, D, B] . [B, C, A] ) [A, C, B] ) So we see that any of the possible cases stays. The case when at least two of the four above points coincide the proof is clear. Deﬁnition 4. The parallel projection between the two lines in the Desargues aﬃne plane, will be called, a function,

Pp : ℓ1 → ℓ2, ∀A, B ∈ ℓ1, AB k Pp(A)Pp(B). It is clear that this function is a bijection between any two lines in Desargues affine planes. Definition 5. [1] A Desargues Affine plane, is said to be ordered, provided (1) All lines in this plane are ordered lines. (2) Parallel projection of the points of one line onto the points of another line in this plane, either preserves or reverses the ordering.

Theorem 4. Each translation (dilation which is different from idP and has no fixed point) in a finite Desargues affine plane AD = (P, L, I) preserves order in a line ℓ of this plane. Proof. Let A, B, C ∈ ℓ, and [A, B, C] . Let’s have, too, a each translation ϕ on this plan. Here we will distinguish two cases regarding the direction of translation. Case.1 The translation ϕ has not direction according to line ℓ, in this case ϕ (ℓ) k ℓ, and ϕ (ℓ) , ℓ, (see Fig. 4). Mark ϕ (ℓ) = ℓ′. From the translation properties, we have,

A, B, C ∈ ℓ =⇒ ϕ (A) ,ϕ (B) ,ϕ (C) ∈ ℓ′. From the translation properties described at [20], we have the following parallelisms,

AB k ϕ (A) ϕ (B) , AC k ϕ (A) ϕ (C) , BC k ϕ (B) ϕ (C) , and

Aϕ (A) k Bϕ (B) k Cϕ (C) . So we have the following parallelograms (see [21]):

A, B,ϕ (B) ,ϕ (A) , A, C,ϕ (C) ,ϕ (A) , B, C,ϕ (C) ,ϕ (B) , ORDEREDLINE&SKEWFIELDINDESARGUESAFFINEPLANE 7

ϕ(A) ϕ(B) ϕ(C) bc bc bc ℓ′

ℓ bc bc bc A B C

′ Figure 4. The translation ϕ as a parallel projection Pp : ℓ → ℓ .

Clearly, we see that the translation ϕ can be seen as parallel projection ′ Aϕ(A) Pp, from line ℓ to line ℓ , with direction the line ℓ . Thus by deﬁnition and so [A, B, C], we have ϕ (A) ,ϕ (B) ,ϕ (C) . Case.2 The translation ϕ has direction according to lineℓ, in this case ϕ (ℓ) k ℓ, and ϕ (ℓ) = ℓ, (see Fig. 5). By translation properties have,

A, B, C ∈ ℓ =⇒ ϕ (A) ,ϕ (B) ,ϕ (C) ∈ ℓ. < E In this case, we choose a point E ℓ of the plan. For this point, ∃!ℓℓ ∈ L, so it exists a translation ϕ1, such that ϕ1 (A) = E, and exists a translation ϕ2, such that ϕ2 (E) = ϕ (A) .

E E = ϕ1(A) ϕ1(B) ϕ1(C) ℓ bc bc bc ℓ ϕ ϕ1 ϕ1 1

ϕ2 ϕ2 ϕ2

ℓ

bc bc bc bc bc bc A B ϕ(A) C ϕ(B) ϕ(C)

Figure 5. The translations ϕ as a composition of two translations ϕ1 and ϕ2 : ϕ = ϕ2 ◦ ϕ1.

Well,

ϕ2 ◦ ϕ1 (A) = ϕ (A) .

Two translations ϕ1and ϕ2 have diﬀerent directions from line ℓ. Now we repeat the ﬁrst case twice, once for the translation ϕ1 and once for the translation ϕ2, and we have: 8 ORGEST ZAKA AND JAMES F. PETERS

[A, B, C] =⇒ ϕ1 (A) ,ϕ1 (B) ,ϕ1 (C) =⇒ ϕ2 ϕ1 (A) ,ϕ2 ϕ1 (B) ,ϕ2 ϕ1 (C) thus,

[A, B, C] =⇒ ϕ2 ◦ ϕ1 (A) , ϕ2 ◦ ϕ1 (B) , ϕ2 ◦ ϕ1 (C) . Hence

[A, B, C] =⇒ ϕ (A) ,ϕ (B) ,ϕ (C) Theorem 5. Every finite ordered Desarguesian affine plane over R2 is Pappian. Proof. Let π be a finite Desarguesian affine plane over R2. From Theorem 3, π is Pappian. Since every line in π satisfies Axioms Lo.1-Lo.4 of definition 1, so every line in π is an ordered line, also, this plane meets the conditions of the definition 5. Consequently, π is a finite ordered Desarguesian affine plane. It has been observed that every Euclidean space is Pappian [17, §1.1, p. 195]. Hence, π is Pappian.

4. The ordered skew-field in a line in ordered desargues affine plane In [23], [4], we have shown how to construct a skew-field over a line in Desargues affine plane. Let it be ℓ a line of Desargues affine plane AD = (P, L, I) . We mark K = (ℓ, +, ∗) the skew-field constructed over the line ℓ, in Desargues affine plane AD. In previous work [23], we have shown how we can transform a line in the Desargues affine plane into an additive Group of its points. We have also shown [4] how to construct a skew-field with a set of points on a line in the Desargues affine plane. In addition, for a line of in any Desargues affine plane, we construct a skew-field with the points of this line, by appropriately defined addition and multiplication of points in a line. During the construction of the skew-field over a line of a Desargues affine plane, we choose two points (each affine plan has at least two points), which we write with O and I and call them zero and one, respectively. These points play the role of unitary elements regarding the two actions addition and multiplication, respectively. Definition 6. [1],[14]A skew-field K is said to be ordered, if, satisfies the following conditions 1. K = K− ∪{0K} ∪ K+ and K− ∩{0K} ∩ K+ = ∅, 2. Forallk1, k2 ∈ K+, k1 + k2 ∈ K+, (K+ is closed under addition) 3. Forallk1, k2 ∈ K+, k1 ∗ k2 ∈ K+, (a product of positive elements is positive) Consider a line ℓ in a Desargues affine plane, by defining the affine plane so that we have at least two points in this line, which we mark O and I. For this line, we have shown in the previous works (see [23],[4]) that we can construct a skew-field K = (ℓ, +, ∗). We have shown [23] the independence of choosing points O and I in a line, for the construction of a skew-field over this line. To establish a separation of the set of points in line ℓ, we separate, firstly the points {O}, which we mark with

K+ = {X ∈ ℓ | [O, X, I] or [O, I, X]} (= ℓ+). ORDEREDLINE&SKEWFIELDINDESARGUESAFFINEPLANE 9 and

K− = {X ∈ ℓ | [X, O, I]} (= ℓ−). So, clearly, from the deﬁnition of the ordered line in a Desargues aﬃne plane, we have

K = K− ∪{0K} ∪ K+, where 0K = O, and

K− ∩{0K} ∩ K+ = ∅.

Lemma 1. For all A, C ∈ K+ =⇒ A + C ∈ K+. Proof. Let’s have the points A, C ∈ ℓ, such that [O, A, C]or [O, C, A]. Suppose we have true [O, A, C] , from the construction of the A + C point we have that A + C ∈ ℓ, (see Fig. 6) which was built according to the algorithm

Step.1.∃B < OI   B A ∀A, C ∈ ℓ,  Step.2.ℓ ∩ ℓ = D  ⇔ A + C = E.  OI OB       Step.3.ℓD ∩ OI = E   CB     

BD B bc bc ℓℓ ℓD ℓOB D OB ℓ ℓ ℓBC ℓBC

ℓ

bc bc bc bc O AC A + C

Figure 6. The addition of points in a line of Desargues aﬃne plane, as a composition of tow parallel projections.

From the construction of the A + C point, we have that:

ℓOB k ℓAD; ℓBC k ℓD(A+C), ℓBD k OI.

From these parallelisms, there is the translation ϕ1 with direction, according to line ℓ, such that, ϕ1(O) = A.

[O, A, C] = O,ϕ1(O), C . Since, by with Theorem 4, the translations preseve the line-order, we have 10 ORGEST ZAKA AND JAMES F. PETERS

[O, A, C] = O,ϕ1(O), C =⇒ ϕ1 (O) ,ϕ1 ϕ1(O) ,ϕ1(C)

=⇒ ϕ1 (O) ,ϕ1 (A) ,ϕ1(C) = [A,ϕ1(A),ϕ1(C)]. Hence, we have deﬁned a translation (see [22],[24]), such that we have true

[O,ϕ1(O) = A,ϕ1(A)] = [O, A,ϕ1(A)] From this translation, we also have,

ϕ1(B) = D and ϕ1(C) = A + C, By by with Theorem 4 we have

[O, A, C] =⇒ ϕ1(O),ϕ1 (A) ,ϕ1(C) = A,ϕ1 (A) , A + C .

For four points O, A,ϕ1 (A) , A + C ∈ ℓ, we have true

[O, A,ϕ1(A)] Lo.3 =⇒ [O, A, A + C] ∧ O,ϕ1(A), A + C . A,ϕ1 (A) , A + C ) By, [O, A, A + C] , we have that

A + C ∈ K+(= ℓ+).

Corollary 1. For all A, C ∈ K− =⇒ A + C ∈ K−.

Corollary 2. For all A ∈ K+=⇒− A ∈ K−.

Proof. If −A ∈ K+ =⇒ A + (−A) = O(= 0K) ∈ K+, which is a contradiction.

Lemma 2. For all A, C ∈ K+ =⇒ A ∗ C ∈ K+. Proof. Let’s have the points A, C ∈ ℓ, such that [O, A, C]or [O, C, A]. Suppose we have true [O, A, C] ,and suppose also that we have [O, I, A] from the construction of the A∗C point we have that A∗C ∈ ℓ, which was built according to the algorithm

Step.1.∃B < OI   A ∀A, C ∈ ℓ,  Step.2.ℓ ∩ OB = E  ⇔ A ∗ C = F.  IB       Step.3.ℓE ∩ OI = F   BC    By, construction of the point A ∗ C, we have parallelisms

IB||AE, BC||E(A ∗ C). IB We take a parallel projection Pp with the direction of the line ℓ , (see Fig. 7) such that

OI OB Pp : ℓ → ℓ , Pp(O) = O; Pp(I) = B

and Pp(A) = E, since, parallel projection, preserve the order, we have

[O, I, A] =⇒ Pp(O), Pp(I), Pp(A) = [O, B, E] IB k AE ) h i ORDERED LINE & SKEW FIELD IN DESARGUES AFFINE PLANE 11

ℓE ℓBC ℓOB Pp E bc e

bc B A ℓℓIB

ℓBC ℓIB ℓOI bc bc bc bc bc I A C A ∗ C O Pp

Figure 7. The multiplication of points in a line of Desargues aﬃne plane, as a composition of tow parallel projections.

But by the multiplication algorithm, during the construction of the A ∗ C point, we have BC||E(A ∗ C). BC We take a parallel projection Pp with the direction of the line ℓ , (see Fig. 7) such that e OB OI Pp : ℓ → ℓ , Pp(O) = O; Pp(B) = C and Pp(E) = A ∗ C. since, parallel projection, preserve the order, we have e e e e [O, B, E] =⇒ Pp(O), Pp(B), Pp(E) = [O, C, A ∗ C] BC||E(A ∗ C) ) h i By, [O, C, A ∗ C] , we have that e e e

A ∗ C ∈ K+(= ℓ+). If we have, [O, A, C] and [A, I, C] =⇒ [O, A, I], (see Fig. 8). By, construction of the point A ∗ C, we have parallelisms

IB||AE, BC||E(A ∗ C). IB We take a parallel projection Pp with the direction of the line ℓ , such that

OI OB Pp : ℓ → ℓ , Pp(O) = O; Pp(A) = E and Pp(I) = B. since, parallel projection, preserve the order, we have

[O, A, I] =⇒ [Pp(O), Pp(A), Pp(I)] =⇒ [O, E, B]. But by the multiplication algorithm, during the construction of the A ∗ C point, we have BC||E(A ∗ C).We take a parallel projection Pp with the direction of the line ℓBC, such that e OB OI Pp : ℓ → ℓ , Pp(O) = O; Pp(E) = A ∗ C and Pp(B) = C. e e e e 12 ORGEST ZAKA AND JAMES F. PETERS

B ℓOB

bc Pp

e IB E bc ℓ ℓBC ℓE ℓBC A ℓℓIB ℓOI bc bc bc bc bc AI A ∗ C C O Pp

Figure 8. The case where, we have, [O, A, C]and [A, I, C].

since, parallel projection, preserve the order, we have

[O, E, B] =⇒ Pp(O), Pp(E), Pp(B) = [O, A ∗ C, C] =⇒ A ∗ C ∈ K+(= ℓ+). h i If we have, [O, A, Ce]and[eA, C, Ie] =⇒ [O, A, I]∧[O, C, I], (see Fig. 9). By, construction of the point A ∗ C, we have parallelisms

IB||AE, BC||E(A ∗ C). IB We take a parallel projection Pp with the direction of the line ℓ , such that

OI OB Pp : ℓ → ℓ , Pp(O) = O; Pp(A) = E and Pp(I) = B. since, parallel projection, preserve the order, we have

[O, A, I] =⇒ [Pp(O), Pp(A), Pp(I)] =⇒ [O, E, B].

P p B ℓE bc ℓOB ℓBC e

E bc BC A ℓ ℓIB ℓℓIB

ℓOI

bc bc bc bc bc O A ∗ C A C I Pp

Figure 9. The case wherewe have, [O, A, C]and [A, C, I]. ORDERED LINE & SKEW FIELD IN DESARGUES AFFINE PLANE 13

But by the multiplication algorithm, during the construction of the A ∗ C point, we have BC||E(A ∗ C).We take a parallel projection Pp with the direction of the line ℓBC, such that e OB OI Pp : ℓ → ℓ , Pp(O) = O; Pp(E) = A ∗ C; Pp(B) = C,

and Pp, since parallele projectione preservese the order, wee have e [O, E, B] =⇒ Pp(O), Pp(E), Pp(B) = [O, A ∗ C, C] =⇒ A ∗ C ∈ K+(= ℓ+). h i e e e We have indirectly from Lemma 2 the case when [O, C, A], since A ∗ C , C ∗ A. From Lemmas 1 and 2, we have the following result.

Theorem 6. Every skew-field that is constructed over an ordered-line of a Desargues affine plane is an ordered skew-field. If we consider the special case where ordered-line of a Desargues affine plane in R2, we obtain the following result.

Corollary 3. Every finite skew-field that is constructed over an ordered-line of a Desargues affine plane in R2 is a finite ordered skew-field. Putting together the result from Corollary 3 and Theorem 5, we obtain the following result for an ordered lines o a finite Desargues affine plane in R2. Theorem 7. A finite skew field constructed over an ordered-line on a finite Desargues affine plane in R2 is Pappian. Proof. Let ℓ be an ordered-line in a finite Desargues affine plane in R2. By definition, ℓ is a Desarguesian affine 1-plane. From Corollary 3, ℓ is a finite ordered skew-field. Hence, from Theorem 5, ℓ is Pappian.

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Orgest ZAKA: Department of Mathematics,Faculty of Technical Science,University of Vlora ’Ismail QEMALI’, Vlora,Albania E-mail address: [email protected], [email protected]

James F. PETERS: Computational Intelligence Laboratory, University of Manitoba, WPG, MB, R3T 5V6, Canada and Department of Mathematics, Faculty of Arts and Sciences, Adiyaman Uni- versity, 02040 Adiyaman,Turkey E-mail address: [email protected]