- Email: [email protected]

Edyta Bartnicka, Metod Saniga

PII:

S0024-3795(19)30446-X

DOI:

https://doi.org/10.1016/j.laa.2019.10.012

Reference:

LAA 15149

To appear in:

Linear Algebra and its Applications

Received date:

13 September 2019

Accepted date:

14 October 2019

Please cite this article as: E. Bartnicka, M. Saniga, Affine and Projective Planes Linked with Projective Lines over Certain Rings of Lower Triangular Matrices, Linear Algebra Appl. (2020), doi: https://doi.org/10.1016/j.laa.2019.10.012.

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Aﬃne and Projective Planes Linked with Projective Lines over Certain Rings of Lower Triangular Matrices Edyta Bartnicka and Metod Saniga Abstract Let Tn (q) be the ring of lower triangular matrices of order n ≥ 2 with entries from the ﬁnite ﬁeld F (q) of order q ≥ 2 and let 2 Tn (q) denote its free left module. For n = 2, 3 it is shown that the projective line over 3(n−1)(n−2) 2 aﬃne planes of order Tn (q) gives rise to a set of (q + 1)(n−1) q q. The points of such an aﬃne plane are non-free cyclic submodules of 2 T (q) not contained in any non-unimodular free cyclic submodule of n 2 T (q) and its lines are points of the projective line. Furthermore, it is n demonstrated that each aﬃne plane can be extended to the projective plane of order q, with the ‘line at inﬁnity’ being represented by those free cyclic submodules of 2 Tn (q) that are generated by non-unimodular pairs. Our approach can be adjusted to address the case of arbitrary n.

Mathematics Subject Classiﬁcation: 15B33, 51E15 Keywords: projective ring lines — rings of lower triangular matrices — non-unimodular free cyclic submodules — aﬃne/projective planes

1

Introduction

The main idea for this paper stems from physics. Some decade ago the second author and his colleagues found out that projective lines over ﬁnite rings have interesting applications in physics, see e. g. [9, 5]. When dealing with lines over rings of order less than 32, it was noticed that in some cases such lines are not only naturally linked with aﬃne planes over certain ﬁnite ﬁelds, but the corresponding rings also feature free cyclic submodules generated by non-unimodular pairs [10]. This paper focuses on projective lines over rings of speciﬁc lower triangular matrices which provide the framework that not only incorporates the most relevant physical cases, but also clariﬁes the so-far unnoticed role played by non-unimodular free cyclic submodules. Through1

out this paper we introduce the family of projective planes built with the set of cyclic submodules of the free left module 2 Tn (q) for n = 2, 3. A special role is played by non-unimodular free cyclic submodules. Constructions of such projective planes are analogous for n = 2 and n = 3, although in the latter case they are technically more diﬃcult. We carry out any such construction in the following steps: 1. We ﬁnd and list all non-unimodular free cyclic submodules of 2 Tn (q) (Theorems 1, 5). 2. We look for all non-free cyclic submodules of 2 Tn (q) not contained in any non-unimodular free cyclic submodule of 2 Tn (q). They are presented as q + 1 pairwise disjoint sets (Propositions 1(b), 3). 3. We completely describe P(Tn (q)). Its points are also presented as q + 1 pairwise disjoint sets (Theorems 2, 6). 4. We show that any submodule from step (2) is contained exclusively in some free cyclic submodules of the respective set/subset points of P(T (q))/P(T3 (q)) (Theorem 2, Proposition 4). 5. We form a point-line incidence structure as follows: its points are submodules from step (2) and its lines are sets of points contained, as submodules, in exactly one free cyclic submodule. This structure is found to be isomorphic to an aﬃne plane of order q (Theorems 3, 7). 6. For any class of parallel lines of such aﬃne plane we choose as a new point a cyclic submodule contained in each line of this class such that all these new submodules are contained in the same non-unimodular free cyclic submodules, which represent a new line. These new points and the new line represent a unique projective closure of the given aﬃne plane (Theorems 4, 8). The results of our paper provide not only a proper understanding of the above-mentioned ’observations’ of physicists, but also point out novel intriguing connections between projective ring lines and aﬃne/projective planes.

2

Preliminaries ⎡

⎢ ⎢ Let Tn (q) be the ring of lower triangular matricesX = ⎢ ⎣

2

0 0 x11 0 x21 x22 0 0 .. .. . . . 0 . . xn1 xn2 . . . xnn

⎤ ⎥ ⎥ ⎥ ⎦

of order n ≥ 2 with entries from the ﬁnite ﬁeld F (q) of order q ≥ 2. For n = 2, i.e., in the case of the ring of ternions, we will use, as usual, the symbol T (q) instead of T2 (q). Let us put F ∗ (q) := F (q)\{0} and let Tn∗ (q) stands for the group of invertible elements of the ring Tn (q). The Jacobson radical of the 2 ring by J. Consider the free left module will be denoted Tn (q) over Tn (q). 2 If X, Y ∈ Tn (q) then the set: Tn (q) X, Y = {A X, Y ; A ∈ Tn (q)} is a 2 left cyclic submodule of If the equation (AX, AY ) = (0, 0) implies Tn (q). that A = 0, then Tn (q) X, Y is free. Remark 1. Obviously, a submodule Tn (q) X, Y ⊂ 2 Tn (q) is free if, and only if, its order is q Sn , where Sn = 1 + 2 + · · · + n. Additionally, such free cyclic submodule is generated exactly by |Tn∗ (q)| elements of the form U (X, Y ), where U runs through all the elements of Tn∗ (q). A pair (X, Y ) ∈ 2 Tn (q) is unimodular, if xii = 0 ∨ yii = 0 for any i = 1, . . . , n. The projective line P(Tn (q)) over the Tn (q) is the set of all ring 2 cyclic submodules Tn (q) X, Y , where X, Y ∈ Tn (q) is unimodular. We refer to [7, 4] for deﬁnitions of the unimodularity and the projective line in the case of an arbitrary ﬁnite associative ring with unity. It is well known that any unimodular pair generates a free cyclic submodule. Therefore the projective line P(Tn (q)) is the set of those free cyclic submodules of 2 Tn (q) which are generated by unimodular pairs. They are called unimodular free cyclic submodules or points of P(Tn (q)). Let us recall some of the most elementary facts of ordinary plane geometry. Deﬁnition 1. An aﬃne plane is a system of points and lines with an incidence relation between the points and lines that satisfy the following axioms. A1 Any two distinct points lie on a unique line. A2 Given any line l and any point P not on l there is a unique line m which contains the point P and does not meet the line l. A3 There exist three non-collinear points. (A set of points P1 , . . . , Pn is said to be collinear if there exists a line l containing them all.) An aﬃne plane is denoted by (A, L), where A is the set of all points and L is the set of all lines. A ﬁnite aﬃne plane (A, L) (with a ﬁnite number of points) is of order n, if there exists a line in L containing exactly n points. We say that two lines of an aﬃne plane (A, L) are parallel if they are equal, or have no point in common. To each set of mutually parallel lines we add a single new point incident with each line of this set. The point added is distinct for each such set. These new points are called points at inﬁnity. We also add a new line, called the line at inﬁnity, incident with all the points at inﬁnity (and no other points). The completion (or closure) (Π, L) of an 3

aﬃne plane (A, L) is deﬁned as follows. The points of (Π, L) are the points of A, plus all the points at inﬁnity of (A, L). A line in (Π, L) is either (a) an ordinary line l of L, plus the point at inﬁnity of l, or (b) the line at inﬁnity, consisting of all the points at inﬁnity of (A, L). The completion (Π, L) is a projective plane (see [3]). A ﬁnite projective plane (Π, L) is of order n, if there exists a line in L containing exactly n + 1 points.

3

Aﬃne and projective planes associated with ternionic projective lines

An important role in our consideration will be played by pairs not contained in any point of the projective line, so called, outliers (as ﬁrst shown in [8], see also [6, 10]). To be more precise, we are interested in those outliers which generate free cyclic submodules. Theorem 1. There are exactly q + 1 free cyclic submodules of 2 T (q) generated by outliers. These are:

1 0 0 0 k 0 1 0 T (q) , , T (q) , , 0 0 1 0 1 0 0 0 where k runs through all the elements of F (q). Proof. In the light of [1, Theorem 1.2] all non-unimodular free cyclic submodules of 2 T (q) are generatd by outliers.

x11 0 y11 0 A non-unimodular pair , ∈ 2 T (q) generates free x21 x22 y21 y22 cyclic submodule of 2 T (q) if, and only if, x22 = y22 = 0 and one of the four conditions is satisﬁed: x11 = 0, y11 = 0, x21 = 0 or x11 = 0, y11 = 0, y21 = 0 or x11 = 0, y11 = 0, x21 = 0, y21 = 0 or x11 = 0, y11 = 0, x21 = 0, y21 = x−1 11 x21 y11 . Hence the total number of outliers generating free cyclic submodules of 2 T (q) is (q − 1)2 (q + 1)q and, according to Remark 1, any such submodule is generated by (q − 1)2 q distinct outliers. Thus the number of non-unimodular free cyclic submodules of 2 T (q) is q + 1. Simple calculations show that the listed non-unimodular free cyclic submodules are pairwise distinct, what completes the proof. Remark 2. A non-free cyclic submodule of 2 T (q) is one of the following forms:

0 0 0 0 1. of order 1: T (q) , ; 0 0 0 0 4

0 0 0 0 , , where p21 , p22 , r21 , r22 ∈ 2. of order q: T (q) r21 r22 p21 p22 F (q) and not all of them are zero;

0 0 r11 0 2 3. of order q : T (q) , with r11 ∈ F ∗ (q), r21 ∈ F (q), r21 0 0 0

p11 0 r11 0 T (q) , , where p11 ∈ F ∗ (q), r11 ∈ F (q), and 0 0 0 0

0 p11 0 0 0 p11 r21 p−1 p11 0 21 T (q) , , , , T (q) 0 p21 0 p21 0 r21 0 0 where p11 , p21 , r21 ∈ F ∗ (q).

Our focus will be on those of order q. Proposition 1. There are two types of cyclic submodules T (q) (P, R) ⊂ T (q) of order q, namely

0 0 0 0 0 0 0 0 (a) with P, R ∈ J: T (q) , , T (q) , , 1 0 0 0 k 0 1 0 with k running through all the elements of F (q); 2

(b) with P ∈ / J ∨R∈ / J; they form q + 1 pairwise disjoint sets: • a single set (hencefort referred to as the ﬁrst set)

0 0 0 0 , , p21 , r21 ∈ F (q) T (q) r21 0 p21 1 and • q disjoint sets (k-sets)

0 0 0 0 T (q) , , p21 , r21 ∈ F (q) , p21 k r21 1 where, as above, k runs through all the elements of F (q). Proof. Let not all p21 , p22 , r21 , r22 ∈ F (q) be equal zero. By Remark 1,

0 0 0 0 0 0 0 0 two pairs , , , generp21 p22 r21 r22 s21 s22 t21 t22 ate the same cyclic submodule if, and only if, there exists u ∈ F ∗ (q) such that s21 = up21 , t21 = up22 , t22 = ur22 . Thus a cyclic submodule = ur21 , s22 0 0 0 0 , of order q is generated by q − 1 distinct T (q) r21 r22 p21 p22 5

0 0 0 0 , , where p21 , r21 ∈ pairs. The number of all pairs r21 0 p21 0 F (q) and p21 = 0∨r21 = 0 is q 2 −1, so the number of all cyclic submodules gen

0 0 0 0 , , erated by them is q+1. The number of all pairs r21 r22 p21 p22 where p21 , p22 , r21 , r22 ∈ F (q) and p22 = 0 ∨ r22 = 0 is q 2 (q 2 − 1), thereby the number of all cyclic submodules generated by them is q 2 (q + 1). It is easy to check that submodules listed in part (a) and submodules in all sets of part (b) are pairwise distinct which leads to the desired claim.

A crucial property of submodules of type (b) is stated in the following result: Proposition 2. Submodules of type (b) are the only non-free cyclic submodules of 2 T (q) not contained in any free cyclic submodule of 2 T (q) generated by an outlier.

y11 0 x11 0 , ⊂ 2 T (q) be free, then Proof. Let a submodule T (q) x21 0 y21 0

a11 x11 0 0 a11 y11 the pair , is equal to the pair a21 x11 + a22 x21 0 a21 y11 + a22 y21 0

0 0 0 0 , if, and only if, a11 = 0, p22 = r22 = 0, p21 = r21 r22 p21 p22 a21 x11 + a22 x21 , r21 = a21 y11 + a22 y21 , thus any submodule of type (b) is not contained in any non-unimodular free cyclic submodule, and any submodule of type (a) is contained in all non-unimodular free cyclic submodules.

0 0 0 0 Of course, T (q) , ⊂ 2 T (q) is contained in all cyclic sub0 0 0 0 modules of 2 T (q). From Remark 2(3) it follows that the number of all pairs generating cyclic submodules of 2 T (q) of order q 2 is q(q 2 −1) and any such submodule is generated by q(q−1) distinct pairs. Hence the number of all distinct cyclic submodules of 2 T (q) of order q 2 is (q+1). By simple calculations we get

1 0 0 0 k 0 1 0 that T (q) , , T (q) , , where k runs 0 0 0 0 0 0 0 0 through all the elements of F (q), are all distinct cyclic submodules of 2 T (q) of order q 2 . For any k ∈ F (q) they are contained in non-unimodular free

1 0 0 0 k 0 1 0 cyclic submodules T (q) , , T (q) , 0 0 1 0 1 0 0 0 of 2 T (q), respectively. It can be shown that these are the only possibilities for submodules of order q 2 to lie in non-unimodular free cyclic submodules of 2 T (q). According to Remark 2 there is no other non-free cyclic submodule of 2 T (q), which completes the proof. 6

Theorem 2. There are exactly q(q +1)2 points of the projective line P(T (q)). They can be presented as q + 1 following sets: the ﬁrst set

y11 0 0 0 1 0 T (q) I, , T (q) , ; x21 , y11 , y21 ∈ F (q) ; y21 0 x21 1 0 0 and q k-sets

x11 0 1 0 0 0 T (q) , I , T (q) ; x11 , x21 , y21 ∈ F (q) , , x21 k 0 k y21 1 where k runs through all the elements of F (q), I denotes the identity matrix. Any submodule of the ﬁrst set of type (b) is contained exclusively in some free cyclic submodules of the ﬁrst set, and any submodule of the k-set of the type (b) is contained exclusively in some free cyclic submodules of the k-set. Proof. According to [2, Corollary 1] there are q(q +1)2 unimodular free cyclic submodules. It is easy to show that the above-listed sets of points of P(T (q)) are pairwise disjoint and each of them contains q 2 + q distinct free cyclic submodules.

x11 0 y11 0 Let , ∈ 2 T (q) be a unimodular pair. Then the x21 x22 y21 y22

0 0 a11 y11 a11 x11 , is equal to pair a21 x11 + a22 x21 a22 x22 a21 y11 + a22 y21 a22 y22

0 0 0 0 the pair , if, and only if, a11 = 0, x22 ∈ F ∗ (q), p21 1 r21 0 −1 −1 −1 y22 = 0, a 22 = x22 , p21 = a21 x11 + x22 x21 , r21 = a21 y11 + x22 y21 , and similarly a11 y11 0 0 a11 x11 , is equal the pair a21 x11 + a22 x21 a22 x22 a21 y11 + a22 y21 a22 y22

0 0 0 0 to the pair , if, and only if, a11 = 0, y22 ∈ F ∗ (q), p21 k r21 1 −1 −1 −1 −1 , k = y22 x22 , p21 = a21 x11 + y22 x21 , r21 = a21 y11 + y22 y21 . a22 = y22 Therefore any submodule of type (b) from a given set is contained in the corresponding set of points of the projective line P(T (q)). Corollary 1. Two cyclic submodules

unimodular free of the form x11 0 y11 0 w11 0 z11 0 T (q) , , T (q) , are x21 x22 y21 y22 w21 w22 z21 z22 in the same set of points of P(T (q)) if, and only if, x22 = w22 , y22 = z22 .

0 0 1 0 1 0 y11 0 , T (q) , Example 1. Let T (q) , y21 0 x21 1 0 0 0 1

0 0 0 0 , be submodules of type be points of P(T (q)), T (q) r21 0 p21 1 7

0 a11 y11 0 0 0 0 a11 0 , = , if, (b). a21 a22 a21 y11 + a22 y21 0 p21 1 r21 0 and only if, a11 = 0, a22 = 1, a21 = p21 , r21 = p21 y11 + y21 for any p21 ∈ F (q),

0 0 0 0 a22 0 0 0 = , if, and only , and a21 0 p21 1 r21 0 a22 x21 a22 if, a11 = 0, a22 = 1, a21 = r21 , p21 = x21 for any r21 ∈ F (q). Hence

0 0 1 0 0 0 y11 0 , ⊂ T (q) , T (q) , y21 0 p21 y11 + y21 0 0 1 p21 1

0 0 0 0 1 0 0 0 , ⊂ T (q) , T (q) r21 0 x21 1 0 0 x21 1

for any p21 , y11 , y21 , x21 , r21 ∈ F (q). By the proof of Proposition 2 these are the only submodules of type (b) contained in the considered points of P(T (q)). Theorem 3. Let us regard cyclic submodules contained in the set

0 0 0 0 T (q) ; p21 , r21 ∈ F (q) , p21 p22 r21 r22 where p22 , r22 are ﬁxed elements of F (q) with p22 = 0 ∨ r22 = 0, as points and free cyclic submodules containing them as lines. These points and lines form a point-line incidence structure isomorphic to the aﬃne plane of order q.

0 0 0 0 Proof. Consider the set T (q) ; p21 , r21 ∈ F (q) . p21 1 r21 0

0 0 0 0 0 0 0 0 Suppose that T (q) , T (q) are p21 1 r21 0 s21 1 t21 0

1 0 y11 0 ∈ P(T (q)). By Example contained in the point T (q) , y21 0 0 1 1 this is equivalent to saying that p21 y11 + y21 = r21 and s21 y11 + y21 = t21 . Consequently (s21 − p21 )y11 = t21 − r21 . If s21 = p21 , then t21 = r21 , so the considered submodules are not distinct. If s21 = p21 , then y11 = (s21 − p21 )−1 (t21 − r21 ). Thus there exists exactly one y11 and exactly one y21 for ﬁxed p21 , r21 , s21 , t21 ∈ F (q).

0 0 0 0 0 0 0 0 , T (q) be conLet now T (q) r21 0 s21 1 t21 0 p21 1

0 0 1 0 tained in the point T (q) , ∈ P(T (q)). By Example 1 x21 1 0 0 this is equivalent to saying that p21 = s21 = x21 . This shows that two distinct submodules of the considered set are contained in exactly one point of 8

P(T (q)). By using results of Example 1 again we get that any submodule of

1 0 y11 0 the considered set not contained in T (q) , ∈ P(T (q)) 0 1 y21 0

0 0 0 0 is of the form T (q) , , where x21 , p21 ∈ F (q), p21 1 p21 y11 + x21 0 x21 = y21 . We get also that elements of the ﬁrst set of points of P(T (q)) which

0 0 0 0 , for any p21 ∈ F (q) are do not contain T (q) p21 y11 + y21 0 p21 1

1 0 y11 0 of the form T (q) , , where y21 ∈ F (q), y21 = y21 . y21 0 1 0

1 0 y11 0 Of course, exactly one of them, i.e., T (q) , ∈ P(T (q)) 0 1 x21 0

0 0 0 0 , ⊂ 2 T (q). contains T (q) p21 y11 + x21 0 p21 1 Let x21 , y21 ∈ F (q) and y21 = x21 . In the same manner we get that there

1 0 0 0 , exists a unique point T (q) ∈ P(T (q)), which contains 0 0 y21 1

0 0 0 0 , ⊂ 2 T (q) and does not contain the submodule T (q) r21 0 y21 1

0 0 0 0 the submodule T (q) , for any r21 ∈ F (q). x21 1 r21 0

x11 0 y11 0 To sum up, given any point T (q) , ∈ P(T (q)) of the x21 1 y21 0

0 0 0 0 ﬁrst set and any submodule T (q) ⊂ 2 T (q), which is p21 1 r21 0 not contained in that point of P(T (q)), there exists a unique point

y11 0 x11 0 , ∈ P(T (q)), which contains that submodule T (q) x21 1 y21 0 and does not contain any submodule of type (b), which is contained in

x11 0 y11 0 T (q) , ∈ P(T (q)). x21 1 y21 0

0 0 0 0 Suppose that the following three submodules: T (q) , , 0 1 0 0

0 0 0 0 0 0 0 0 T (q) , , T (q) , are contained in a 0 1 1 0 1 1 0 0

y11 0 x11 0 , ∈ P(T (q)). Then there free cyclic submodule T (q) x21 1 y21 0 9

⎧ 1. a21 x11 + x21 = 0. ⎪ ⎪ ⎪ ⎪ ⎪ 2. a21 y11 + y21 = 0. ⎪ ⎪ ⎪ ⎨3. b x + x = 0. 21 11 21 exist a21 , b21 , c21 ∈ F (q) such that ⎪4. b21 y11 + y21 = 1. ⎪ ⎪ ⎪ ⎪ ⎪ 5. c21 x11 + x21 = 1. ⎪ ⎪ ⎩ 6. c21 y11 + y21 = 0. Thus (b21 − a21 )x11 = 0, (b21 − a21 )y11 = 1. Hence b21 = a21 , x11 = 0 and x21 = −a21 x11 = 0. But then c21 x11 + x21 = 0, which contradicts equation (5) and thereby it contradicts the assumption that the three above submodules are contained in the same point of P(T (q)). It means that there exist

0 0 0 0 three submodules T (q) ⊂ 2 T (q) not contained in the p21 1 r21 0 same point of P(T (q)). Moreover, any point P(T (q)) of the ﬁrst set contains

0 0 0 0 ⊂ 2 T (q), what follows exactly q submodules T (q) r21 0 p21 1 from Example 1. We have just shown that all axioms of an aﬃne plane are met for the considered set. By the same methods it follows that submodules of type (b) from a given k-set and points of P(T (q)) containing them give an incidence structure isomorphic to the aﬃne plane of order q. So, there are altogether q + 1 isomorphic aﬃne planes of order q linked with P(T (q)). Corollary 1. Two free cyclic submodules generated unimodular by pairs

2. y11 0 w11 0 z11 0 x11 0 , , , represent x21 x22 y21 y22 w21 w22 z21 z22 lines of the same aﬃne plane associated with P(T (q)) if, and only if, x22 = w22 , y22 = z22 . 2. Two submodules generated of the form

unimodular free cyclic

by pairs x11 0 y11 0 w11 0 z11 0 , , , represent x21 x22 y21 y22 w21 x22 z21 y22 the same class of parallel lines of an aﬃne plane associated with P(T (q)) if, and only if, x11 = w11 , y11 = z11 . 3. Two submodules generated of the form free cyclic

by pairs

unimodular y11 0 x11 0 y11 0 x11 0 , , , represent x21 x22 y21 y22 w21 x22 z21 y22 distinct parallel lines of an aﬃne plane associated with P(T (q)) if, and only if, x21 = w21 or y21 = z21 . This is, however, not a full story as it turns out that any of these aﬃne plane 10

can be extended to the projective plane of order q due to the fact that in 2 T (q) there also exist non-unimodular pairs generating free cyclic submodules. Theorem 4. Any aﬃne plane of order q associated with P(T (q)) can be extended to the projective plane of order q in the following way. Consider the set free cyclic submodules generated by pairs

of unimodular x11 0 y11 0 , ∈ 2 T (q), such that x22 , y22 ∈ F (q) are ﬁxed. x21 x22 y21 y22 1. For all such submodules, where x11 , y11 ∈ F (q) are also ﬁxed, a submod

0 0 0 0 , ⊂ 2 T (q) must be taken into account ule T (q) y11 0 x11 0 as a new point, and

x22 0 y22 0 2. A free cyclic submodule T (q) , ⊂ 2 T (q) must y22 0 δy22 0 0 be taken into account as a new line (δy22 0 denotes the Kronecker delta). Points of the obtained projective plane are all cyclic submodules of order q of 2 T (q) contained in a given set of points of P(T (q)), and lines are these points of P(T (q)) and the respective non-unimodular free cyclic submodule.

y11 0 x11 0 , ∈ P(T (q)), where x22 , y22 ∈ Proof. Points T (q) x21 x22 y21 y22 F (q) are ﬁxed, represent all lines of an aﬃne plane associated with P(T (q)).

0 0 0 0 , ⊂ 2 T (q) is contained in 1. A submodule T (q) y11 0 x11 0

x11 0 y11 0 each submodule T (q) , ∈ P(T (q)), which, x21 x22 y21 y22 by Corollary 2 (2), represents a line of a given parallel class, and is distinct for each such class of an aﬃne plane associated with P(T (q)).

y22 0 x22 0 , ⊂ 2 T (q) 2. By Proposition 2 and its proof T (q) y22 0 δy22 0 0

0 0 0 0 contains T (q) , ⊂ 2 T (q) for all x11 , y11 ∈ F (q) x11 0 y11 0 and it does not contain any submodule of type (b), i.e. a new line (represented by a non-unimodular free cyclic submodule) is incident with all new points (represented by submodules of type (a)) and no other points.

11

Summing up, cyclic submodules of order q generated by pairs with both entries in J are the points and any non-unimodular free cyclic submodule incorporating these submodules is the line representing the unique projective closure of all aﬃne planes associated with P(T (q)). Figure 1 serves as an illustration of our ﬁndings for the case q = 2. 10

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Figure 1: A visualisation of all free cyclic submodules over T (2) after [10]. The 18 free cyclic submodules generated by unimodular pairs are represented by colored broken polygons, whereas those three generated by non-unimodular vectors are represented by black curves. For the sake of simplicity, the matrices of T (2) are labeled by integers from 0 to 7 in such a way that J = {0, 6} (for more details, see [8, 10]). The three aﬃne planes are represented by the following set of points: {T (2)3, 0 , T (2)3, 6 , T (2)5, 0 , T (2)5, 6 }, {T (2) 3, 3 , T (2) 3, 5 , T (2) 5, 3 , T (2) 5, 5 }’ {T (2) 6, 3 , T (2) 6, 5 , T (2) 0, 3 , T (2) 0, 5 } and two parallel lines have the same color; the closure line comprises the submodules T (2) 6, 0 , T (2) 6, 6 and T (2) 0, 6 . 12

4

Aﬃne and projective planes associated with P(T3(q))

Theorem 5. There are exactly (q + 1)2 (2q 2 + q + 1) free cyclic submodules of 2 T3 (q) generated by outliers. They can be presented as q + 1 sets: the ﬁrst set and q k-sets, where k ∈ F (q). The ﬁrst set consists of submodules generated by pairs: ⎛⎡ ⎤⎞ ⎛⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎞ 1 0 0 1 0 0 0 0 0 0 0 0 ⎝⎣ 0 0 0 ⎦ , ⎣ 1 0 0 ⎦⎠ , ⎝⎣ 0 0 0 ⎦ , ⎣ 1 0 0 ⎦⎠ , 0 x32 1 0 x32 0 0 y32 y33 0 y32 1 ⎛⎡ ⎤ ⎡ ⎤⎞ ⎛⎡ ⎤ ⎡ ⎤⎞ 1 0 0 0 0 0 1 0 0 0 0 0 ⎝⎣ 0 0 0 ⎦ , ⎣ 1 0 0 ⎦⎠ , ⎝⎣ 0 0 0 ⎦ , ⎣ 1 0 0 ⎦⎠ , 0 x32 0 0 1 0 0 1 0 0 0 0 ⎤⎞ ⎤ ⎞ ⎛⎡ ⎤ ⎡ ⎛⎡ ⎤ ⎡ 1 0 0 0 0 0 1 0 0 0 0 0 ⎝⎣ 0 1 0 ⎦ , ⎣ y21 y22 0 ⎦⎠ , ⎝⎣ 0 1 0 ⎦ , ⎣ y21 y22 0 ⎦⎠ , y31 1 0 1 0 0 0 0 0 0 0 0 ⎤ ⎡ ⎛⎡ ⎤ ⎡ ⎤⎞ ⎛⎡ ⎤⎞ 0 0 0 0 0 0 1 0 0 1 0 0 ⎝⎣ 0 0 0 ⎦ , ⎣ y21 1 0 ⎦⎠ , ⎝⎣ 0 0 0 ⎦ , ⎣ y21 1 0 ⎦⎠ , 0 0 0 0 1 0 y31 0 0 1 0 0 where x32 , y21 , y22 , y31 , y32 , y33 ∈ F (q); Each of k-sets consists of submodules generated by pairs: ⎛⎡

⎤ ⎡ k 0 0 1 ⎝⎣ 1 0 0 ⎦ , ⎣ 0 0 x32 x33 0 ⎛⎡ ⎤ ⎡ k 0 0 1 ⎝⎣ 1 0 0 ⎦ , ⎣ 0 0 1 0 0 ⎛⎡ ⎤ ⎡ k 0 0 1 ⎝⎣ x21 x22 0 ⎦ , ⎣ 0 x31 1 0 0 ⎛⎡ ⎤ ⎡ k 0 0 1 ⎝⎣ x21 1 0 ⎦ , ⎣ 0 0 x31 0 0

⎤⎞ ⎛⎡ k 0 0 0 0 ⎦⎠ , ⎝⎣ 1 0 y32 1 ⎤⎞ ⎛⎡ k 0 0 ⎦ ⎠ ⎝ ⎣ , 1 0 0 0 y32 0 ⎤⎞ ⎛⎡ 0 0 k 1 0 ⎦⎠ , ⎝⎣ x21 0 0 0 0 0 0 1 0

⎤ ⎡ ⎤⎞ 0 0 1 0 0 0 0 ⎦ , ⎣ 0 0 0 ⎦⎠ , x32 1 0 y32 0 ⎤ ⎡ ⎤⎞ 0 0 1 0 0 0 0 ⎦ , ⎣ 0 0 0 ⎦⎠ , 0 0 0 1 0 ⎤ ⎡ ⎤⎞ 1 0 0 0 0 x22 0 ⎦ , ⎣ 0 1 0 ⎦⎠ , 1 0 0 0 0 0 ⎤ ⎞ ⎛⎡ ⎤ ⎡ ⎤⎞ k 0 0 1 0 0 ⎦⎠ , ⎝⎣ x21 1 0 ⎦ , ⎣ 0 0 0 ⎦⎠ , 1 0 0 0 0 0 13

where x21 , x22 , x31 , x32 , x33 , y32 ∈ F (q). Proof. In the light of [1, Theorem 1.2] all non-unimodular free cyclic submodules of 2 T3 (q) are generatd by outliers. ⎛⎡ ⎤ ⎡ ⎤⎞ x11 0 0 y11 0 0 A non-unimodular pair ⎝⎣ x21 x22 0 ⎦ , ⎣ y21 y22 0 ⎦⎠ ∈ 2 T3 (q) x31 x32 x33 y31 y32 y33 generates free cyclic submodule of 2 T3 (q) if, and only if, x22 = y22 = 0 or x22 = y22 = 0 and one of the following conditions is satisﬁed: 1. x11 = 0, y11 = 0 and (a) x22 = y22 = 0, x21 = 0, x32 = 0 ∨ x33 = 0 ∨ y32 = 0 ∨ y33 = 0, or −1 (b) x22 = 0, y22 = 0, x31 = x21 x−1 22 x32 ∨ (x31 = x21 y22 y32 ∧ x32 = 0), or −1 (c) x22 = 0, y22 = 0, x31 = x21 x−1 22 x32 ∨ (x31 = x21 x22 x32 ∧ y32 = 0), or −1 −1 (d) x22 = 0, y22 = 0, x32 = x22 y22 y32 ∨ (x32 = x22 y22 y32 ∧ x31 = −1 x21 y22 y32 ).

2. x11 = 0, y11 = 0 and (a) x22 = y22 = 0, y21 = 0, x32 = 0 ∨ x33 = 0 ∨ y32 = 0 ∨ y33 = 0, or −1 −1 y32 ∨ (y31 = y21 y22 y32 ∧ x32 = 0), or (b) x22 = 0, y22 = 0, y31 = y21 y22 −1 (c) x22 = 0, y22 = 0, y31 = y21 x−1 22 x32 ∨ (y31 = y21 x22 x32 ∧ y32 = 0), or −1 −1 y32 ∨ (x32 = x22 y22 y32 ∧ x31 = (d) x22 = 0, y22 = 0, x32 = x22 y22 −1 x21 y22 y32 ),

3. x11 = 0, y11 = 0 and −1 y21 , or (a) x22 = y22 = 0, x21 = x11 y11 −1 y31 + (x21 − (b) x22 = 0, y22 = 0, x32 = 0 ∨ x32 = 0 ∧ x31 = x11 y11 −1 −1 x11 y11 y21 )y22 y32 , or −1 y31 + (x21 − (c) x22 = 0, y22 = 0, y32 = 0 ∨ y32 = 0 ∧ x31 = x11 y11 −1 −1 x11 y11 y21 )x22 x32 , or −1 −1 (d) x22 = 0, y22 = 0, x32 = x22 y22 y32 ∨ x32 = x22 y22 y32 ∧ x31 = −1 −1 −1 x11 y11 y31 + (x21 − x11 y11 y21 )y22 y32 .

Hence the total number of outliers generating free cyclic submodules of 2 T3 (q) is (q − 1)3 q 3 (q + 1)2 (2q 2 + q + 1). Remark 1 implies that any such submodule is generated by (q − 1)3 q 3 distinct outliers. So the number of non-unimodular free cyclic submodules of 2 T3 (q) is (q + 1)2 (2q 2 + q + 1). 14

By using Remark 1 again and multiplying outliers by all invertible matrices of T3∗ (q) from the left we get immediately that the listed non-unimodular free cyclic submodules are pairwise distinct. Just as in the case of ternions, we are interested in non-free cyclic submodules not contained in any free cyclic submodule generated by an outlier. Proposition 3. There are exactly (q + 1)2 q 5 non-free cyclic submodules of 2 T3 (q) not contained in any non-unimodular free cyclic submodule of 2 T3 (q). They can be presented as q + 1 sets having q 2 + q subsets each. The ﬁrst set consists of q 2 subsets of the form ⎧ ⎫ ⎛⎡ ⎤ ⎡ ⎤⎞ 0 0 0 0 0 0 ⎨ ⎬ T3 (q) ⎝⎣ p21 1 0 ⎦ , ⎣ r21 0 0 ⎦⎠ ; p21 , p31 , r21 , r31 ∈ F (q) , ⎩ ⎭ p31 0 1 r31 r32 r33 where r32 , r33 run through all the elements of F (q), and q subsets of the form ⎧ ⎫ ⎛⎡ ⎤ ⎡ ⎤⎞ 0 0 0 0 0 0 ⎨ ⎬ T3 (q) ⎝⎣ p21 1 0 ⎦ , ⎣ r21 0 0 ⎦⎠ ; p21 , p31 , r21 , r31 ∈ F (q) , ⎩ ⎭ p31 0 0 r31 r32 1 where r32 runs through all the elements of F (q). Each of q k-sets, where k ∈ F (q), consists of q 2 subsets of the form ⎫ ⎧ ⎤ ⎡ ⎛⎡ ⎤⎞ 0 0 0 0 0 0 ⎬ ⎨ T3 (q) ⎝⎣ p21 k 0 ⎦ , ⎣ r21 1 0 ⎦⎠ ; p21 , p31 , r21 , r31 ∈ F (q) , ⎭ ⎩ p31 p32 p33 r31 0 1 where p32 , p33 run through all the elements of F (q), and q subsets of the form ⎫ ⎧ ⎛⎡ ⎤ ⎡ ⎤⎞ 0 0 0 0 0 0 ⎬ ⎨ ⎝ ⎣ ⎦ ⎣ ⎦ ⎠ ; p21 , p31 , r21 , r31 ∈ F (q) , T3 (q) p21 k 0 , r21 1 0 ⎭ ⎩ p31 p32 1 r31 0 0 where p32 runs through all the elements of F (q). Proof. It is easy to show that subsets speciﬁed in the assertion are pairwise 4 disjoint and any of them contains 2 q distinct pairs. Suppose now that T3 (q) X, Y ⊂ T3 (q) is a free cyclic submodule containing ⎤⎞ ⎤ ⎡ ⎛⎡ 0 0 0 0 0 0 a submodule generated by a pair ⎝⎣ p21 1 0 ⎦ , ⎣ r21 0 0 ⎦⎠. Then p31 0 1 r31 r32 r33 15

x11 = 0 ∨ y11 = 0 and there exists A ∈ T3 (q) such that ⎡

⎤ ⎡ ⎤ 0 0 a11 x11 0 0 0 ⎣ ⎦ = ⎣ p21 1 0 ⎦ , a21 x11 + a22 x21 a22 x22 0 a31 x11 + a32 x21 + a33 x31 a32 x22 + a33 x32 a33 x33 p31 0 1 ⎤ ⎡ ⎤ 0 0 0 0 0 a11 z11 ⎦ = ⎣ r21 0 0 ⎦ . ⎣ a21 y11 + a22 y21 a22 y22 0 a31 y11 + a32 y21 + a33 y31 a32 y22 + a33 y32 a33 y33 r31 r32 r33 Hence x22 = 0 and x33 = 0, so the pair X, Y is unimodular. By the same methods it follows that all remaining speciﬁed submodules are not contained in any non-unimodular free cyclic submodule. We will show now that any other non-free cyclic submodules of 2 T3 (q) is contained in some free cyclic submodule generated by an outlier. Of course, p22 = r22 = 0 or p33 = r33 = ⎛⎡ ⎤ ⎡ ⎤⎞ p11 0 0 r11 0 0 0 for all pairs (P, R) = ⎝⎣ p21 p22 0 ⎦ , ⎣ r21 r22 0 ⎦⎠ ∈ 2 T3 (q) p31 p32 p33 r31 r32 r33 generating these non-free cyclic submodules. Below we consider all the cases of such pairs (P, R), giving also the matrix A ∈ T3 (q) and the outlier (X, Y ) ∈ 2 T3 (q) generating a free cyclic submodule such that A(X, Y ) = (P, R). 1. p11 = 0; 1.1 p22 = r22 = 0; 1.1.1. p32 = 0 ∨ p33 = 0 ∨ r32 = 0 ∨ r33 = 0, r21 = p21 p−1 11 r11 ; ⎡ ⎤ p11 0 0 ⎣ A = p21 0 0 ⎦, p31 −p31 p−1 11 r11 + r31 1 ⎤ ⎡ −1 ⎤⎞ ⎛⎡ p11 r11 0 0 1 0 0 (X, Y ) = ⎝⎣ 0 0 0 ⎦ , ⎣ 1 0 0 ⎦⎠ ; 0 r32 r33 0 p32 p33 1.1.2. p32 = p33 = r32 = r33 = 0; ⎡ ⎤ p11 0 0 ⎦, A = ⎣ p21 −p21 p−1 11 r11 + r21 0 −1 p31 −r31 p11 r11 + r31 0 ⎤⎞ ⎛⎡ ⎤ ⎡ −1 1 0 0 p11 r11 0 0 (X, Y ) = ⎝⎣ 0 0 0 ⎦ , ⎣ 1 0 0 ⎦⎠ ; 0 0 0 0 1 0 ⎡

−1 −1 1.2. p22 = 0, p33 = y33 = 0, r31 = p31 p−1 11 r11 + p32 p22 (r21 − p21 p11 r11 ), r32 = p32 p−1 22 r22 ;

16

⎡

⎤ p11 0 0 A = ⎣ p21 1 0 ⎦, −1 p31 p32 p22 0 ⎤ ⎡ ⎤⎞ ⎛⎡ p−1 0 0 1 0 0 11 r11 ⎦⎠ ; (X, Y ) = ⎝⎣ 0 p22 0 ⎦ , ⎣ −p21 p−1 11 r11 + r21 r22 0 0 0 0 0 1 0 −1 1.3. p22 = 0, r22 = 0, p33 = r33 = 0, p32 = 0, r31 = p31 p−1 11 r11 + r32 r22 (r21 − −1 −p21 p11 r11 ); ⎡ ⎤ p11 0 0 A = ⎣ p21 1 0 ⎦, −1 p31 r32 r22 0 ⎛⎡ ⎤ ⎡ ⎤⎞ 0 0 1 0 0 p−1 11 r11 ⎦⎠ ; (X, Y ) = ⎝⎣ 0 0 0 ⎦ , ⎣ −p21 p−1 11 r11 + r21 r22 0 0 1 0 0 0 0 2. p11 = 0, r11 = 0; 2.1 p22 = r22 = 0; 2.1.1. p32 = 0 ∨ p33 = 0 ∨ r32 = 0 ∨ r33 = 0, p21 = 0; ⎡ ⎤ ⎡ ⎤⎞ ⎤ ⎛⎡ r11 0 0 1 0 0 0 0 0 A = ⎣ r21 0 0 ⎦, (X, Y ) = ⎝⎣ 1 0 0 ⎦ , ⎣ 0 0 0 ⎦⎠ ; r31 p31 1 0 r32 r33 0 p32 p33 2.1.2. p32 = p33 = r32 = r33 = 0; ⎡ ⎤ ⎛⎡ ⎤ ⎡ ⎤⎞ r11 0 0 0 0 0 1 0 0 A = ⎣ r21 p21 0 ⎦, (X, Y ) = ⎝⎣ 1 0 0 ⎦ , ⎣ 0 0 0 ⎦⎠ ; r31 p31 0 0 0 0 0 1 0 −1 2.2. p22 = 0, p33 = r33 = 0, p31 = p32 p−1 22 p21 , r32 = p32 p22 r22 ; ⎡ ⎤ ⎤ ⎡ ⎤⎞ ⎛⎡ r11 0 0 0 0 0 1 0 0 A = ⎣ r21 1 0 ⎦, (X, Y ) = ⎝⎣ p21 p22 0 ⎦ , ⎣ 0 r22 0 ⎦⎠ ; −1 r31 p32 p22 0 0 0 0 0 1 0 −1 p21 ; 2.3. p22 = 0, r22 = 0, p33 = r33 = 0, p32 = 0, p31 = r32 r22 ⎤ ⎡ ⎤ ⎛⎡ ⎡ 0 0 1 0 0 0 0 r11 ⎦ ⎣ ⎦ ⎝ ⎣ ⎣ A = r21 1 0 , (X, Y ) = p21 0 0 , 0 r22 −1 r31 r32 r22 0 0 1 0 0 0 3. p11 = r11 = 0; 3.1 p22 = r22 = 0; 3.1.1. p32 = 0 ∨ p33 = 0 ∨ r32 = 0 ∨ r33 = 0; ⎡ ⎤ ⎡ ⎤ ⎛⎡ 0 0 0 0 0 1 0 0 ⎣ ⎦ ⎣ ⎦ ⎝ ⎣ A = p21 r21 0 , (X, Y ) = , 1 0 0 0 0 p31 0 1 r31 r32 0 p32 p33

17

⎤⎞ 0 0 ⎦⎠ ; 0 ⎤⎞ 0 0 ⎦⎠ ; r33

3.1.2. p32 = p33 ⎡ 0 0 A = ⎣ p21 r21 p31 r31

= r32 = r33 = 0; ⎤ ⎛⎡ ⎤ ⎡ ⎤⎞ 0 1 0 0 0 0 0 0 ⎦, (X, Y ) = ⎝⎣ 0 0 0 ⎦ , ⎣ 1 0 0 ⎦⎠ ; 0 0 1 0 0 0 0

−1 0, p33 = r33 = 0; 3.2.1. r31 = p32 p−1 3.2. p22 = 22 r21 ∨ r32 = p32 p22 r22 ; ⎡ ⎤ 0 0 0 A = ⎣ p21 1 0 ⎦, −1 p31 p32 p22 1 ⎤ ⎡ ⎤⎞ ⎛⎡ 0 0 0 1 0 0 (X, Y ) = ⎝⎣ 0 p22 0 ⎦ , ⎣ r21 r22 0 ⎦⎠. −1 0 0 0 −p32 p−1 22 r21 + r31 −p32 p22 r22 + r32 0 −1 3.2.2. r31 = p32 p−1 22 r21 , r32 = p32 p22 r22 ; ⎛⎡ ⎡ ⎤ ⎤ ⎡ ⎤⎞ 1 0 0 0 0 0 0 0 0 A = ⎣ p21 1 0 ⎦, (X, Y ) = ⎝⎣ 0 p22 0 ⎦ , ⎣ r21 r22 0 ⎦⎠; 0 p31 p32 p−1 0 0 0 0 1 0 22 −1 r21 ; 3.3. p22 = 0, r22 = 0, p33 = r33 = 0; 3.3.1. p32 = 0 ∨ r31 = r32 r22 ⎤ ⎡ 0 0 0 ⎣ A = p21 1 0 ⎦, −1 p31 r32 r22 1 ⎤ ⎡ ⎤⎞ ⎛⎡ 0 0 0 1 0 0 (X, Y ) = ⎝⎣ 0 0 0 ⎦ , ⎣ r22 0 ⎦⎠. r21 −1 −r32 r22 r21 + r31 0 0 0 p32 0 −1 r21 ; 3.3.2. p32 = 0, r31 = r32 r22 ⎤⎞ ⎡ ⎤ ⎛⎡ ⎤ ⎡ 0 0 0 1 0 0 0 0 0 A = ⎣ p21 1 0 ⎦, (X, Y ) = ⎝⎣ 0 0 0 ⎦ , ⎣ r21 r22 0 ⎦⎠. −1 p31 r32 r22 0 0 0 0 0 1 0

Theorem 6. There are exactly (q+1)3 q 3 points of the projective line P(T3 (q)). They can be presented as q + 1 sets consists of q 2 + q subsets each: • a single set, referred to as the ﬁrst set, consists of: – q 2 subsets of the form T3 (q) X , Y , T3 (q) X, Y ; x21 , x31 , y11 , y21 , y31 ∈ F (q) , where ⎤⎞ ⎤ ⎡ y11 0 0 1 0 0 X , Y = ⎝⎣ 0 1 0 ⎦ , ⎣ y21 0 0 ⎦⎠ , 0 0 1 y31 y32 y33 ⎛⎡

18

⎤⎞ ⎤ ⎡ 1 0 0 0 0 0 X, Y = ⎝⎣ x21 1 0 ⎦ , ⎣ 0 0 0 ⎦⎠ x31 0 1 0 y32 y33 ⎛⎡

and y32 , y33 run through all the elements of F (q); – q subsets of the form T3 (q) X , Y , T3 (q) X, Y ; x21 , x31 , y11 , y21 , y31 ∈ F (q) , where ⎛⎡

1 0 0 X , Y = ⎝⎣ 0 1 0 0 0 0 ⎛⎡ 0 0 X, Y = ⎝⎣ x21 1 x31 0

⎤ ⎡

y11 ⎦ , ⎣ y21 y31 ⎤ ⎡ 1 0 0 ⎦,⎣ 0 0 0

⎤⎞ 0 0 0 0 ⎦⎠ , y32 1 ⎤⎞ 0 0 0 0 ⎦⎠ y32 1

and y32 runs through all the elements of F (q). • q k-sets, where k ∈ F (q); each of them consists of: – q 2 subsets of the form T3 (q) X, Y , T3 (q) X , Y ; x11 , x21 , x31 , y21 , y31 ∈ F (q) , where

X, Y

⎤ ⎡ ⎤⎞ 1 0 0 x11 0 0 = ⎝⎣ x21 k 0 ⎦ , ⎣ 0 1 0 ⎦⎠ , x31 x32 x33 0 0 1 ⎛⎡

⎤ ⎡ ⎤⎞ 0 0 0 1 0 0 X , Y = ⎝⎣ 0 k 0 ⎦ , ⎣ y21 1 0 ⎦⎠ y31 0 1 0 x32 x33

⎛⎡

and x32 , x33 run through all the elements of F (q); – q subsets of the form T3 (q) X, Y , T3 (q) X , Y ; x11 , x21 , x31 , y21 , y31 ∈ F (q) , where

X, Y

⎤ ⎡ ⎤⎞ 1 0 0 x11 0 0 = ⎝⎣ x21 k 0 ⎦ , ⎣ 0 1 0 ⎦⎠ , 0 0 0 x31 x32 1 ⎛⎡

19

⎛⎡

⎤ ⎡ ⎤⎞ 1 0 0 0 0 0 X , Y = ⎝⎣ 0 k 0 ⎦ , ⎣ y21 1 0 ⎦⎠ y31 0 0 0 x32 1

and x32 runs through all the elements of F (q). Proof. The number (q+1)3 q 3 of unimodular free cyclic submodules of P(T3 (q)) follows directly from [2, Corollary 1]. Let Tn (q)(X, Y ), Tn (q)(W, Z) ∈ P(Tn (q)), then Remark 1 implies that uii xii = wii ∧ uii yii = zii . Tn (q)(X, Y ) = Tn (q)(W, Z) ⇒ xii ,yii ∈F ∗ (q); uii ∈F ∗ (q) i=1,...,n

Furthermore, u22 k = l and u22 1 = 1 if, and only if, u22 = 1 and k = l. These two facts exclude the possibility that points of P(T3 (q)) in two distinct subsets are the same. Thus the listed (q+1)2 q subsets of points of P(T3 (q)) are pairwise disjoint. Moreover, by using Remark 1 again we get that two points T3 (q)(X, Y ), T3 (q)(W, Z) ∈ P(T3 (q)) of the same subset are equal if, and only if, they are proportional by the identity matrix, equivalently (X, Y ) = (W, Z). So, each of the listed subsets contains q 2 (q + 1) distinct points. Corollary 3. Two unimodular free cyclic ⎤ ⎡ ⎛⎡ y11 x11 0 0 ⎦ ⎣ ⎝ ⎣ , y21 T3 (q) x21 x22 0 x31 x32 x33 y31 ⎤ ⎡ ⎛⎡ z11 w11 0 0 ⎦ ⎣ ⎝ ⎣ , z21 T3 (q) w21 w22 0 w31 w32 w33 z31

submodules ⎤⎞ 0 0 y22 0 ⎦⎠ ∈ P(T3 (q)), y32 y33 ⎤⎞ 0 0 z22 0 ⎦⎠ ∈ P(T3 (q)) z32 z33

are in the same set of points of P(T3 (q)) if, and only if, w22 = x22 , z22 = y22 . Two unimodular free cyclic submodules ⎤ ⎡ ⎤⎞ ⎛⎡ y11 0 0 x11 0 0 T3 (q) ⎝⎣ x21 x22 0 ⎦ , ⎣ y21 y22 0 ⎦⎠ ∈ P(T3 (q)), x31 x32 x33 y31 y32 y33 ⎤ ⎡ ⎤⎞ ⎛⎡ z11 0 0 w11 0 0 T3 (q) ⎝⎣ w21 x22 0 ⎦ , ⎣ z21 y22 0 ⎦⎠ ∈ P(T3 (q)) w31 w32 w33 z31 z32 z33 are in the same subset of points of P(T3 (q)) if, and only if, w32 = x32 , w33 = x33 , z32 = y32 , z33 = y33 . 20

Proposition 4. Let

⎛⎡

⎤ ⎡ ⎤⎞ 0 0 0 0 0 0 T3 (q) P, R = T3 (q) ⎝⎣ p21 p22 0 ⎦ , ⎣ r21 r22 0 ⎦⎠ ⊂ 2 T3 (q), p31 p32 p33 r31 r32 r33

be a non-free cyclic submodule not contained in any non-unimodular free cyclic submodule of 2 T3 (q), and let ⎤ ⎡ ⎤⎞ ⎛⎡ y11 0 0 x11 0 0 T3 (q) X, Y = T3 (q) ⎝⎣ x21 x22 0 ⎦ , ⎣ y21 y22 0 ⎦⎠ ∈ P(T3 (q)), x31 x32 x33 y31 y32 y33 where X, Y is of the form as in Theorem 6. T3 (q) P, R ⊂ T3 (q) X, Y if, and only if, all of the following conditions are met: 1. x22 = p22 , x32 = p32 , x33 = p33 , y22 = r22 , y32 = r32 , y33 = r33 ; 2. x21 = p21 − r21 x11 , x31 = p31 − r31 x11 if y11 = 1, y21 = y31 = 0; 3. y21 = r21 − p21 y11 , y31 = r31 − p31 y11 if x11 = 1, x21 = x31 = 0. 2 pair. Proof. Let (X, Y ) ∈ T3 (q) be a unimodular Suppose that T3 (q) P, R ⊂ T3 (q) X, Y . Equivalently, there exists a matrix ⎤ ⎡ a11 0 0 A = ⎣ a21 a22 0 ⎦ ∈ T3 (q) such that A(X, Y ) = (P, R) . a31 a32 a33 ⎤⎞ ⎛⎡ ⎤ ⎡ 0 0 0 0 0 0 If T3 (q) P, R = T3 (q) ⎝⎣ p21 1 0 ⎦ , ⎣ r21 0 0 ⎦⎠, then y22 = 0, p31 0 1 r31 r32 r33 −1 −1 ∗ x22 , x33 ∈ F (q) and a22 = x22 , a33 = x33 . By Theorem 6 we can put on x22 = x33 = 1, x32 = 0. Therefore r32 = a32 y22 + a33 y32 = y32 , r33 = a33 y33 = y33 . In the same manner it can be shown that x22 = p22 , x32 = p32 , x33 = p33 , y22 = r22 , y32 = r32 , y33 = r33 in all other cases of non-free cyclic submodules of 2 2 T3 (q) not contained in any non-unimodular free cyclic submodule of T3 (q). In the light of Theorem 6 we can consider two cases of T3 (q) X, Y :

1. x11 = 1, x21 = x31 = 0, then A(X, Y ) = (P, R) ⇔ a11 = a32 = 0, a22 = a33 = 1, a21 = p21 , a31 = p31 , y21 = r21 − p21 y11 , y31 = r31 − p31 y11 ; 2. y11 = 1, y21 = y31 = 0, then A(X, Y ) = (P, R) ⇔ a11 = a32 = 0, a22 = a33 = 1, a21 = r21 , a31 = r31 , x21 = p21 − r21 x11 , x31 = p31 − r31 x11 .

21

Corollary 4. Let ⎤ ⎡ ⎤⎞ 0 0 0 0 0 0 T3 (q) P, R = T3 (q) ⎝⎣ p21 p22 0 ⎦ , ⎣ r21 r22 0 ⎦⎠ ⊂ 2 T3 (q), p31 p32 p33 r31 r32 r33

⎛⎡

be a non-free cyclic submodule not contained in any non-unimodular free cyclic submodule of 2 T3 (q). ⎤ ⎡ ⎤⎞ ⎛⎡ y11 0 0 x11 0 0 Two elements T3 (q) X, Y = T3 (q) ⎝⎣ x21 x22 0 ⎦ , ⎣ y21 y22 0 ⎦⎠ , x31 x32 x33 y31 y32 y33 ⎛⎡ ⎤ ⎡ ⎤⎞ x11 0 0 y11 0 0 T3 (q) X , Y = T3 (q) ⎝⎣ x21 x22 0 ⎦ , ⎣ y21 y22 0 ⎦⎠ of the ﬁxed y32 y33 x31 x32 x33 y 31 subset of points of P(T3 (q)), where X, Y , X , Y are of the form as in Theorem 6, have no submodule T3 (q) P, R in common if, and only if, ⎡

1 0 • X = X = ⎣ 0 x22 0 x32 y31 = y31 , or ⎡ 1 0 ⎣ • Y = Y = 0 y22 0 y32 x31 = x31 .

⎤ ⎤ ⎡ 0 y11 0 0 = y21 ∨ 0 ⎦ , Y = ⎣ y21 y22 0 ⎦, and y21 y31 y32 y33 x33 ⎤ ⎤ ⎡ 0 x11 0 0 0 ⎦ , X = ⎣ x21 x22 0 ⎦, and x21 = x21 ∨ x31 x32 x33 y33

⎤ a11 0 0 Example 2. Let y32 , y33 be ﬁxed elements of F (q) and A = ⎣ a21 a22 0 ⎦ a31 a32 a33 be a matrix of T3 (q). Choose the subset of points of P(T3 (q)) of the form T3 (q) X , Y , T3 (q) X, Y ; x21 , x31 , y11 , y21 , y31 ∈ F (q) , where ⎡

⎛⎡ ⎤ ⎡ 1 0 0 y11 0 X , Y = ⎝⎣ 0 1 0 ⎦ , ⎣ y21 0 0 0 1 y31 y32 ⎤ ⎡ ⎛⎡ 1 0 0 0 0 X, Y = ⎝⎣ x21 1 0 ⎦ , ⎣ 0 0 x31 0 1 0 y32 22

⎤⎞ 0 0 ⎦⎠ , y33 ⎤⎞ 0 0 ⎦⎠ . y33

⎤⎞ ⎤ ⎡ 0 0 0 0 0 0 A X , Y = ⎝⎣ p21 1 0 ⎦ , ⎣ r21 0 0 ⎦⎠ ⇔ a11 = a32 = 0, a22 = p31 0 1 r31 y32 y33 = a33 = 1, a21 = p21 , a31 = p31 , r21 = p21 y11 + y21 , r31 = p31 y11 + y31 and p21 , p31 ∈ F (q). ⎛⎡ ⎤⎞ ⎤ ⎡ 0 0 0 0 0 0 A X, Y = ⎝⎣ p21 1 0 ⎦ , ⎣ r21 0 0 ⎦⎠ ⇔ a11 = a32 = 0, a22 = p31 0 1 r31 y32 y33 = a33 = 1, a21 = r21 , a31 = r31 , p21 = x21 , p31 = x31 and r21 , r31 ∈ F (q). Hence ⎛⎡ ⎤⎞ ⎤ ⎡ 0 0 0 0 0 0 T3 (q) ⎝⎣ p21 1 0 ⎦ , ⎣ p21 y11 + y21 0 0 ⎦⎠ ⊂ 2 T3 (q) X , Y , p31 0 1 p31 y11 + y31 y32 y33 ⎛⎡

⎤⎞ ⎤ ⎡ 0 0 0 0 0 0 T3 (q) ⎝⎣ x21 1 0 ⎦ , ⎣ r21 0 0 ⎦⎠ ⊂ 2 T3 (q) X, Y x31 0 1 r31 y32 y33 ⎛⎡

for any p21 , p31 , y11 , y21 , y31 , y32 , y33 , x21 , x31 , r21 , r31 , y32 , y33 ∈ F (q). According to Proposition 4 these are the only non-free cyclic submodules of 2 T3 (q) not contained in any non-unimodular free cyclic submodule of 2 T3 (q) and contained in the considered points of P(T (q)). Theorem 7. Let ⎧ ⎫ ⎤ ⎡ ⎤⎞ ⎛⎡ 0 0 0 0 0 0 ⎨ ⎬ A = T3 (q) ⎝⎣ p21 x22 0 ⎦ , ⎣ r21 y22 0 ⎦⎠ ; p31 , r31 ∈ F (q) , ⎩ ⎭ p31 x32 x33 r31 y32 y33 where p21 , x22 , x32 , x33 , r21 , y22 , y32 , y33 are ﬁxed elements of F (q), be contained in a subset of non-free cyclic submodules not contained in any non-unimodular free cyclic submodules, and let us regard its elements as points and free cyclic submodules containing them as lines. These points and lines form a point-line incidence structure isomorphic to the aﬃne plane of order q. ⎤ ⎡ ⎤⎞ ⎛⎡ 0 0 0 0 0 0 Proof. Let T3 (q) P, R = T3 (q) ⎝⎣ p21 x22 0 ⎦ , ⎣ r21 y22 0 ⎦⎠ and p31 x32 x33 r31 y32 y33 ⎤ ⎡ ⎤⎞ ⎛⎡ 0 0 0 0 0 0 T3 (q) S, V = T3 (q) ⎝⎣ s21 x22 0 ⎦ , ⎣ v21 y22 0 ⎦⎠ be any two s31 x32 x33 v31 y32 y33 23

distinct submodules of A. Assume that they both are contained in the ⎛⎡ ⎤ ⎡ ⎤⎞ 1 0 0 y11 0 0 point T3 (q) ⎝⎣ 0 x22 0 ⎦ , ⎣ y21 y22 0 ⎦⎠ ∈ P(T3 (q)). By Propo0 x32 x33 y31 y32 y33 sition 4 we obtain: r21 = p21 y11 + y21 , r31 = p31 y11 + y31 , v21 = s21 y11 + y21 , v31 = s31 y11 + y31 . Hence (r31 − v31 ) = (p31 − s31 )y11 . p31 = s31 , so y11 = (p31 − s31 )−1 (r31 − v31 ). Therefore, y11 is uniquely determined and so there are y21 , y31 . Similarly, we get that x11 , x21 , x31 are also uniquely determined in case of submodules T3 (q)(P, R), T3 (q)(S, V ) contained in the point ⎛⎡ ⎤ ⎡ ⎤⎞ x11 0 0 1 0 0 T3 (q)(X, Y ) = T3 (q) ⎝⎣ x21 x22 0 ⎦ , ⎣ 0 y22 0 ⎦⎠ ∈ P(T3 (q)). x31 x32 x33 0 y32 y33 We have just shown that any two distinct submodules of A are contained in exactly one point of P(T3 (q)). Thereby axiom A1 of Deﬁnition 1 is satisﬁed. ⎤ ⎡ ⎤⎞ ⎛⎡ y11 0 0 1 0 0 Let T3 (q)(X, Y ) = T3 (q) ⎝⎣ 0 x22 0 ⎦ , ⎣ y21 y22 0 ⎦⎠ be a ﬁxed 0 x32 x33 y31 y32 y33 element of the set L of all free cyclic submodules containing elements of A. From Proposition 4 we get: • an element of A not contained in ⎛⎡ 0 0 ⎝ ⎣ T3 (q)(P, R ) = T3 (q) p21 x22 p31 x32

the point T3 (q)(X, Y ) is of the form ⎤ ⎡ ⎤⎞ 0 0 0 0 0 ⎦ , ⎣ p21 y11 + y21 y22 0 ⎦⎠ x33 y32 y33 p31 y11 + y31

where y31 ∈ F (q) and y31 = y31 ;

• a point of L not containing any element of A contained in T3 (q)(X, Y ) is of the form ⎤ ⎡ ⎤⎞ ⎛⎡ y11 0 0 1 0 0 T3 (q)(X, Y ) = T3 (q) ⎝⎣ 0 x22 0 ⎦ , ⎣ y21 y22 0 ⎦⎠ , 0 x32 x33 y31 y32 y33 ∈ F (q) and y31 = y31 ; where y31

• there is a unique point of the form T3 (q)(X, Y ), which contains a ﬁxed submodule T3 (q)(P, R ), namely ⎤ ⎡ ⎤⎞ ⎛⎡ 1 0 0 y11 0 0 T3 (q)(X, Y ) = T3 (q) ⎝⎣ 0 x22 0 ⎦ , ⎣ y21 y22 0 ⎦⎠ . y32 y33 0 x32 x33 y31 24

We have just proved that axiom A2 of Deﬁnition 1 is satisﬁed for any ⎛⎡

⎤ ⎡ ⎤⎞ 1 0 0 y11 0 0 T3 (q) ⎝⎣ 0 x22 0 ⎦ , ⎣ y21 y22 0 ⎦⎠ ∈ P(T3 (q)). 0 x32 x33 y31 y32 y33 Of course, the result is the same for any free cyclic submodule ⎤ ⎡ ⎤⎞ 1 0 0 x11 0 0 T3 (q) ⎝⎣ x21 x22 0 ⎦ , ⎣ 0 y22 0 ⎦⎠ ∈ P(T3 (q)). x31 x32 x33 0 y32 y33 ⎤ ⎡ ⎤ ⎡ 0 0 0 0 0 0 Assume that submodules T3 (q) ⎣ 0 x22 0 ⎦ , ⎣ 0 y22 0 ⎦ of p31 x32 x33 r31 y32 y33 any set A such that p31 = 0, r31 = 0 (the ﬁrst one), p31 = 0, r31 = 1 (the second one), and p31 = 1, r31 = 0 (the third one) are contained in the same free ⎤ ⎡ ⎤⎞ ⎛⎡ y11 0 0 x11 0 0 cyclic submodule T3 (q) ⎝⎣ x21 x22 0 ⎦ , ⎣ y21 y22 0 ⎦⎠ ∈ P(T3 (q)). x31 x32 x33 y y y ⎧ 31 32 33 1. a31 x11 + x31 = 0. ⎪ ⎪ ⎪ ⎪ ⎪ 2. a31 y11 + y31 = 0. ⎪ ⎪ ⎪ ⎨3. b x + x = 0. 31 11 31 Then there exist a31 , b31 , c31 ∈ F (q) such that Hence ⎪4. b31 y11 + y31 = 1. ⎪ ⎪ ⎪ ⎪ ⎪ 5. c31 x11 + x31 = 1. ⎪ ⎪ ⎩ 6. c31 y11 + y31 = 0. (b31 − a31 )x11 = 0, (b31 − a31 )y11 = 1. Consequently b31 = a31 , x11 = 0 and x31 = −a31 x11 = 0. But then we get c31 x11 + x31 = 0, which contradicts equation (5) and thereby it contradicts the assumption that the above-given three submodules are contained in the same point of P(T3 (q)). It means that for any set A there exist three submodules not contained in the same point of P(T3 (q)), thereby axiom A3 of Deﬁnition 1 is satisﬁed. Moreover, any element of L contains exactly q submodules of A, what follows directly from Proposition 4. This shows that any set A and the points of P(T3 (q)) containing elements of A give a point-line incidence structure isomorphic to the aﬃne plane of order q. So, there are altogether q 3 (q+1)2 mutually isomorphic aﬃne planes of order q associated with the projective line P(T3 (q)). ⎛⎡

The above considerations lead to the following results:

25

1. Two unimodular free cyclic submodules ⎤ ⎡ ⎤⎞ ⎛⎡ y11 0 0 x11 0 0 T3 (q) ⎝⎣ x21 x22 0 ⎦ , ⎣ y21 y22 0 ⎦⎠ , x31 x32 x33 y31 y32 y33

Corollary 5.

⎤ ⎡ ⎤⎞ z11 0 0 w11 0 0 T3 (q) ⎝⎣ w21 w22 0 ⎦ , ⎣ z21 z22 0 ⎦⎠ w31 w32 w33 z31 z32 z33 ⎛⎡

of P(T3 (q)) represent lines of aﬃne planes of the same subset if, and only if, w22 = x22 , w32 = x32 , w33 = x33 , z22 = y22 , z32 = y32 , z33 = y33 . This follows from Corollary 3 and Theorem 7. 2. Two unimodular free cyclic submodules ⎤ ⎡ ⎤⎞ ⎛⎡ y11 0 0 x11 0 0 T3 (q) X, Y = T3 (q) ⎝⎣ x21 x22 0 ⎦ , ⎣ y21 y22 0 ⎦⎠ , x31 x32 x33 y31 y32 y33 ⎤ ⎡ ⎤⎞ z11 0 0 w11 0 0 T3 (q) W, Z = T3 (q) ⎝⎣ w21 x22 0 ⎦ , ⎣ z21 y22 0 ⎦⎠ w31 x32 x33 z31 y32 y33 of P(T3 (q)), where X, Y , W, Z are of the form as in Theorem 6, represent lines of the same aﬃne plane if, and only if, there exist nonfree cyclic submodules ⎤ ⎡ ⎤⎞ ⎛⎡ 0 0 0 0 0 0 T3 (q) ⎝⎣ s21 x22 0 ⎦ , ⎣ s21 y22 0 ⎦⎠ , s31 x32 x33 s31 y32 y33

⎛⎡

⎤ ⎡ ⎤⎞ 0 0 0 0 0 0 T3 (q) ⎝⎣ s21 x22 0 ⎦ , ⎣ s21 y22 0 ⎦⎠ y32 y33 v31 x32 x33 v31 ⎛⎡

of 2 T3 (q) not contained in any non-unimodular free cyclic submodule of 2 T3 (q) such that: x21 = p21 − r21 x11 , x31 = p31 − r31 x11 if y11 = 1, y21 = y31 = 0, y21 = r21 − p21 y11 , y31 = r31 − p31 y11 if x11 = 1, x21 = x31 = 0, w21 = p21 − r21 w11 , w31 = s31 − v31 w11 if z11 = 1, z21 = z31 = 0, z21 = r21 − p21 z11 , z31 = v31 − s31 z11 if w11 = 1, w21 = w31 = 0. This follows from Proposition 4 and Theorem 7. 26

3. Two unimodular free cyclic submodules ⎤ ⎡ ⎛⎡ y11 x11 0 0 T3 (q) X, Y = T3 (q) ⎝⎣ x21 x22 0 ⎦ , ⎣ y21 x31 x32 x33 y31 ⎤ ⎡ ⎛⎡ z11 w11 0 0 T3 (q) W, Z = T3 (q) ⎝⎣ w21 x22 0 ⎦ , ⎣ z21 w31 x32 x33 z31 of P(T3 (q)) represent parallel lines of an aﬃne plane ⎤ ⎡ ⎛⎡ x11 0 0 y11 ⎝ ⎣ ⎦ ⎣ T3 (q) W, Z = T3 (q) , y21 x21 x22 0 w31 x32 x33 z31

⎤⎞ 0 0 y22 0 ⎦⎠ , y32 y33 ⎤⎞ 0 0 y22 0 ⎦⎠ y32 y33

if, and only if, ⎤⎞ 0 0 y22 0 ⎦⎠ . y32 y33

They are distinct exactly if, w31 = x31 or z31 = y31 . This follows from Corollary 4 and Theorem 7. Theorem 8. Any aﬃne plane of order q associated with P(T3 (q)) can be extended to the projective plane of order q in the following way. Consider the set of all submodules ⎤ ⎡ ⎤⎞ ⎛⎡ y11 0 0 x11 0 0 T3 (q) ⎝⎣ x21 x22 0 ⎦ , ⎣ y21 y22 0 ⎦⎠ ∈ P(T3 (q)) x31 x32 x33 y31 y32 y33 representing lines of a given aﬃne plane associated with P(T3 (q)). 1. For all such submodules, where x11 , x21 , y11 , y21 are ﬁxed elements of F (q), that is to say for ﬁxed set of parallel lines of an aﬃne plane, a submodule ⎛⎡ ⎤ ⎡ ⎤⎞ 0 0 0 0 0 0 T3 (q) ⎝⎣ 0 0 0 ⎦ , ⎣ 0 0 0 ⎦⎠ ⊂ 2 T3 (q) x11 0 0 y11 0 0 must be taken into account as a new point, and 2. A free cyclic submodule ⎛⎡ ⎤ ⎡ ⎤⎞ x22 0 0 0 0 y22 T3 (q) ⎝⎣ y22 0 0 ⎦ , ⎣ δy22 0 0 0 ⎦⎠ ⊂ 2 T3 (q), 0 δy22 0 0 0 y22 0 where δy22 0 stands for the Kronecker delta, must be taken into account as a new line. 27

⎤ ⎡ ⎤⎞ y11 0 0 x11 0 0 Proof. Obviously, T3 (q) ⎝⎣ x21 x22 0 ⎦ , ⎣ y21 y22 0 ⎦⎠ ∈ P(T3 (q)) x31 x32 x33 y31 y32 y33 ⎛⎡ ⎤ ⎡ ⎤⎞ 0 0 0 0 0 0 ⎝ ⎣ ⎦ ⎣ contains T3 (q) 0 0 0 , 0 0 0 ⎦⎠. By Corollary 5 (2 and 3) x11 0 0 y11 0 0 we get immediately that if unimodular free cyclic submodules ⎤ ⎡ ⎤⎞ ⎛⎡ y11 0 0 x11 0 0 T3 (q) ⎝⎣ x21 x22 0 ⎦ , ⎣ y21 y22 0 ⎦⎠ , x31 x32 x33 y31 y32 y33 ⎛⎡

⎤ ⎡ ⎤⎞ z11 0 0 w11 0 0 T3 (q) ⎝⎣ w21 x22 0 ⎦ , ⎣ z21 y22 0 ⎦⎠ w31 x32 x33 z31 y32 y33 ⎛⎡

represent non-parallel lines of an aﬃne plane, then x11 = w11 or y11 = z11 . According to Theorem 6 we can assume that x11 = w11 = 1 or y11 = z11 = 1. ⎛⎡ ⎤ ⎡ ⎤⎞ 0 0 0 0 0 0 Consequently each of submodules T3 (q) ⎝⎣ 0 0 0 ⎦ , ⎣ 0 0 0 ⎦⎠, x11 0 0 y11 0 0 ⎛⎡ ⎤ ⎡ ⎤⎞ 0 0 0 0 0 0 ⎝ ⎣ ⎦ ⎣ T3 (q) 0 0 0 , 0 0 0 ⎦⎠ of 2 T3 (q) contained in distinct sets w11 0 0 z11 0 0 of parallel lines of an aﬃne plane meets one of the two conditions: x11 = w11 = 1, y11 = z11 or y11 = z11 = 1, x11 = w11 , so they are also distinct. We have shown that any line of a set of parallel lines of an aﬃne plane contains the added new point and these new points are distinct for distinct sets of parallel lines. By Theorem 5 it suﬃces to consider two cases of non-unimodular free ⎛⎡ ⎤ ⎡ ⎤⎞ 0 0 0 0 x22 y22 cyclic submodules T3 (q) ⎝⎣ y22 0 0 ⎦ , ⎣ δy22 0 0 0 ⎦⎠ ⊂ 2 T3 (q), 0 δy22 0 0 0 y22 0 namely x22 = 1, y22 = 0 and x22 ∈ F (q), y22 = 1. It is easy to verify now that for any aﬃne plane the added new line contains all the new points. Theorem 9. Let us regard sets of parallel lines of aﬃne planes associated with a subset of points of P(T3 (q)) as lines, and sets of points of such aﬃne planes as points. These points and lines form a point-line incidence structure, called a 2-aﬃne plane, isomorphic to the aﬃne plane of order q. A point and a line of a 2-aﬃne plane are incident if any element of the set representing this point is contained in some element of the set representing this line. 28

Proof. According to Corollary 5 (3) and Theorem 6, sets of parallel lines of aﬃne planes associated with a subset of points of P(T3 (q)), i.e., lines of a 2-aﬃne plane, are one of the two following forms: ⎧ ⎫ ⎛⎡ ⎤ ⎡ ⎤⎞ 1 0 0 y11 0 0 ⎨ ⎬ T3 (q) ⎝⎣ 0 x22 0 ⎦ , ⎣ y21 y22 0 ⎦⎠ ; y31 ∈ F (q) , ⎩ ⎭ 0 x32 x33 y31 y32 y33 ⎫ ⎤ ⎡ ⎤⎞ 1 0 0 0 0 0 ⎬ T3 (q) ⎝⎣ x21 x22 0 ⎦ , ⎣ 0 y22 0 ⎦⎠ ; x31 ∈ F (q) , ⎭ ⎩ x31 x32 x33 0 y32 y33 ⎧ ⎨

⎛⎡

where x21 , y11 , y21 run through all the elelments of F (q), x22 = 1, y22 = 0 and x32 , x33 , y32 , y33 are ﬁxed elements of F (q) such that x33 = 1 ∨ y33 = 1; or ⎧ ⎫ ⎤ ⎡ ⎤⎞ ⎛⎡ 1 0 0 x11 0 0 ⎨ ⎬ T3 (q) ⎝⎣ x21 x22 0 ⎦ , ⎣ 0 y22 0 ⎦⎠ ; x31 ∈ F (q) , ⎩ ⎭ x31 x32 x33 0 y32 y33 ⎧ ⎨

⎫ ⎤ ⎡ ⎤⎞ 1 0 0 0 0 0 ⎬ ⎝ ⎣ ⎦ ⎦ ⎠ ⎣ T3 (q) , y21 y22 0 ; y31 ∈ F (q) , 0 x22 0 ⎩ ⎭ 0 x32 x33 y31 y32 y33 ⎛⎡

where x11 , x21 , y21 run through all the elelments of F (q), y22 = 1 and x22 , x32 , x33 , y32 , y33 are ﬁxed elements of F (q) such that x33 = 1 ∨ y33 = 1. Then sets of points of such aﬃne planes, i.e., points of a 2-aﬃne plane, are of the form ⎧ ⎫ ⎤ ⎡ ⎤⎞ ⎛⎡ 0 0 0 0 0 0 ⎨ ⎬ T3 (q) ⎝⎣ p21 x22 0 ⎦ , ⎣ r21 y22 0 ⎦⎠ ; p31 , r31 ∈ F (q) , ⎩ ⎭ p31 x32 x33 r31 y32 y33 where p21 , r21 run through all the elelments of Consider two distinct points ⎧ ⎤ ⎡ ⎛⎡ 0 0 0 0 0 ⎨ ⎦ ⎣ ⎝ ⎣ P = T3 (q) , r21 y22 p21 x22 0 ⎩ p31 x32 x33 r31 y32 ⎧ ⎨

F (q). ⎫ ⎤⎞ 0 ⎬ 0 ⎦⎠ ; p31 , r31 ∈ F (q) , ⎭ y33

⎫ ⎤ ⎡ ⎤⎞ 0 0 0 0 0 0 ⎬ S = T3 (q) ⎝⎣ s21 x22 0 ⎦ , ⎣ v21 y22 0 ⎦⎠ ; s31 , v31 ∈ F (q) . ⎩ ⎭ s31 x32 x33 v31 y32 y33 ⎛⎡

29

By Proposition 4 they are inicident with a line l if, and only if, ⎧ ⎨

⎫ ⎤ ⎡ ⎤⎞ y11 0 0 1 0 0 ⎬ ⎦ ⎣ ⎦ ⎠ ⎝ ⎣ l = l1 = T3 (q) , y21 y22 0 ; y31 ∈ F (q) , 0 x22 0 ⎩ ⎭ 0 x32 x33 y31 y32 y33 ⎛⎡

where y21 = r21 − p21 y11 = v21 − s21 y11 , then r21 − v21 = (p21 − s21 )y11 ; or ⎧ ⎫ ⎤ ⎡ ⎤⎞ ⎛⎡ 1 0 0 x11 0 0 ⎨ ⎬ ⎦ ⎣ ⎦ ⎠ ⎝ ⎣ l = l2 = T3 (q) , 0 y22 0 ; x31 ∈ F (q) , x21 x22 0 ⎩ ⎭ x31 x32 x33 0 y32 y33 where x21 = p21 −r21 x11 = s21 −v21 x11 ; hence, we get p21 −s21 = (r21 −v21 )x11 . P = S ⇔ p21 = s21 ∨ r21 = v21 , thus we can consider three following cases: 1. p21 = s21 , r21 = v21 , then ⎧ ⎛⎡ 0 ⎨ l = l2 = T3 (q) ⎝⎣ p21 ⎩ x31

there exists a unique line ⎫ ⎤ ⎡ ⎤⎞ 0 0 1 0 0 ⎬ x22 0 ⎦ , ⎣ 0 y22 0 ⎦⎠ ; x31 ∈ F (q) , ⎭ 0 y32 y33 x32 x33

incident with P and S. 2. p21 = s21 , r21 = v21 , then there exists a unique ⎧ ⎤ ⎡ ⎛⎡ 0 0 1 0 0 ⎨ ⎦ ⎣ ⎝ ⎣ , r21 y22 l = l1 = T3 (q) 0 x22 0 ⎩ 0 x32 x33 y31 y32

line ⎫ ⎤⎞ 0 ⎬ 0 ⎦⎠ ; y31 ∈ F (q) , ⎭ y33

incident with P and S. 3. p21 = s21 , r21 = v21 , then l1 consists of unimodular free cyclic submodules of P(T3 (q)) generated by pairs ⎛⎡ ⎤ ⎡ ⎤⎞ 1 0 0 0 0 (p21 − s21 )−1 (r21 − v21 ) ⎝⎣ 0 x22 0 ⎦ , ⎣ r21 − p21 (p21 − s21 )−1 (r21 − v21 ) y22 0 ⎦⎠ , 0 x32 x33 y31 y32 y33 where y31 runs through all the elements of F (q), and l2 consists of unimodular free cyclic submodules of P(T3 (q)) generated by pairs ⎛⎡ ⎤ ⎡ ⎤⎞ (r21 − v21 )−1 (p21 − s21 ) 0 0 1 0 0 ⎝⎣ p21 − r21 (r21 − v21 )−1 (p21 − s21 ) x22 0 ⎦ , ⎣ 0 y22 0 ⎦⎠ , x32 x33 x31 0 y32 y33 30

where x31 runs through all the elements of F (q). By using Proposition 4 again we get that l1 consists of unimodular free cyclic submodules of P(T3 (q)) generated by pairs: ⎛⎡ ⎤ ⎡ ⎤⎞ 0 0 1 0 0 (p21 − s21 )−1 (r21 − v21 ) ⎝⎣ 0 x22 0 ⎦ , ⎣ r21 − p21 (p21 − s21 )−1 (r21 − v21 ) y22 0 ⎦⎠ , 0 x32 x33 r31 − p31 (p21 − s21 )−1 (r21 − v21 ) y32 y33 where p31 , r31 run through all the elements of F (q), and l2 consists of unimodular free cyclic submodules of P(T3 (q)) generated by pairs: ⎛⎡ ⎤ ⎡ ⎤⎞ 0 0 (r21 − v21 )−1 (p21 − s21 ) 1 0 0 ⎝⎣ p21 − r21 (r21 − v21 )−1 (p21 − s21 ) x22 0 ⎦ , ⎣ 0 y22 0 ⎦⎠ , p31 − r31 (r21 − v21 )−1 (p21 − s21 ) x32 x33 0 y32 y33 where p31 , r31 runs through all the elements of F (q). Let (X1 , Y1 ), (X2 , Y2 ) be pairs generating element of l1 and l2 , respec⎤ ⎡ 0 0 (r21 − v21 )−1 (p21 − s21 ) tively. Matrix A = ⎣ p21 − r21 (r21 − v21 )−1 (p21 − s21 ) 1 0 ⎦ leads to p31 − r31 (r21 − v21 )−1 (p21 − s21 ) 0 1 the equation A(X1 , Y1 ) = (X2 , Y2 ) for any pair (X1 , Y1 ), and matrix ⎡ ⎤ (p21 − s21 )−1 (r21 − v21 ) 0 0 B = ⎣ r21 − p21 (p21 − s21 )−1 (r21 − v21 ) 1 0 ⎦ leads to the equation r31 − p31 (p21 − s21 )−1 (r21 − v21 ) 0 1 B(X2 , Y2 ) = (X1 , Y1 ) for any pair (X2 , Y2 ). So, by the Remark 1, l1 = l2 ; consequently, there exists a unique line incident with P and S. Thereby ⎧ axiom A1 of an aﬃne plane is proven. ⎫ ⎤ ⎡ ⎤⎞ ⎛⎡ y11 0 0 1 0 0 ⎨ ⎬ Let l = T3 (q) ⎝⎣ 0 x22 0 ⎦ , ⎣ y21 y22 0 ⎦⎠ ; y31 ∈ F (q) , be a ⎩ ⎭ 0 x32 x33 y31 y32 y33 ﬁxed set of parallel lines of an aﬃne plane associated with a subset C of P(T3 (q)). From Proposition 4 we get: • a set of points of any aﬃne plane associated with C such that their elements are not contained in any element of l is of the form ⎧ ⎫ ⎤ ⎡ ⎤⎞ ⎛⎡ 0 0 0 0 0 0 ⎨ ⎬ P = T3 (q) ⎝⎣ p21 x22 0 ⎦ , ⎣ r21 y22 0 ⎦⎠ ; p31 , r31 ∈ F (q) , ⎩ ⎭ p31 x32 x33 r31 y32 y33 where r21 = y21 + p21 y11 ; 31

• a set l of parallel lines of an aﬃne plane associated with C such that any element of l has nothing in common with any submodule not contained in a non-unimodular free cyclic submodule and contained in some element of l, is of the form ⎧ ⎫ ⎤ ⎡ ⎤⎞ ⎛⎡ 1 0 0 y11 0 0 ⎨ ⎬ ⎝ ⎣ ⎦ ⎣ ⎦ ⎠ l = T3 (q) , y21 y22 0 ; y31 ∈ F (q) , 0 x22 0 ⎩ ⎭ 0 x32 x33 y31 y32 y33 = y21 ; where y21

• any element of the ﬁxed set P is contained in some element of l if, and only if, y21 = r21 − p21 y11 . So, for any line l and any point P not on l there is a unique line l which contains the point P and does not meet the line l. The result is the same in the case of a set ⎧ ⎫ ⎤ ⎡ ⎤⎞ ⎛⎡ x11 0 0 1 0 0 ⎨ ⎬ l = T3 (q) ⎝⎣ x21 x22 0 ⎦ , ⎣ 0 y22 0 ⎦⎠ ; x31 ∈ F (q) , ⎩ ⎭ x31 x32 x33 0 y32 y33 of parallel lines of an aﬃne plane associated an aﬃne plane is satisﬁed. Assume that elements of sets ⎧ ⎤ ⎡ ⎛⎡ 0 0 0 0 0 ⎨ ⎦ ⎣ ⎝ ⎣ , T3 (q) 0 y22 0 x22 0 ⎩ p31 x32 x33 r31 y32

with P(T3 (q)). So, axiom A2 of ⎫ ⎤⎞ 0 ⎬ 0 ⎦⎠ ; p31 , r31 ∈ F (q) , ⎭ y33

⎫ ⎤ ⎡ ⎤⎞ 0 0 0 0 0 0 ⎬ T3 (q) ⎝⎣ 0 x22 0 ⎦ , ⎣ 1 y22 0 ⎦⎠ ; p31 , r31 ∈ F (q) ⎭ ⎩ p31 x32 x33 r31 y32 y33 ⎧ ⎨

⎛⎡

of points of aﬃne planes associated with subset C of P(T3 (q)) are contained in elements of the same set l of parallel lines of an aﬃne plane associated with C. By Proposition 4 we get immediately that ⎧ ⎫ ⎤ ⎡ ⎤⎞ ⎛⎡ 1 0 0 0 0 0 ⎨ ⎬ ⎦ ⎣ ⎦ ⎠ ⎝ ⎣ l = T3 (q) , 0 y22 0 ; x31 ∈ F (q) 0 x22 0 ⎩ ⎭ x31 x32 x33 0 y32 y33

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and elements of the set ⎧ ⎫ ⎤ ⎡ ⎤⎞ ⎛⎡ 0 0 0 0 0 0 ⎨ ⎬ T3 (q) ⎝⎣ 1 x22 0 ⎦ , ⎣ 0 y22 0 ⎦⎠ ; p31 , r31 ∈ F (q) ⎩ ⎭ p31 x32 x33 r31 y32 y33 of points of an aﬃne plane associated with C are not contained in elements of l. It means that axiom A3 of an aﬃne plane is satisﬁed. Moreover, elements of any set of parallel lines of an aﬃne plane associated with subset C of P(T3 (q)) contain q sets of points of aﬃne planes associated with C, what follows directly from Proposition 4. So, there are altogether q(q + 1)2 mutually isomorphic 2-aﬃne planes of order q. Figure 2 serves as a visualization of the structure of a 2-aﬃne plane of order three.

Figure 2: A pictorial representation of the aﬃne plane of order three (left) and its 2-aﬃne counterpart (right).

5

Concluding remarks

Although we studied in detail only the case of n = 2 and n = 3, it represents no diﬃculty for the interested reader to readily adjust our lines of reasoning 33

to the case of arbitrary n. In the future we also plan to analyse other similar types/classes of ﬁnite associative rings with unity, in particular those exhibiting non-unimodular free cyclic submodules whose ‘homomorphic’ images are projective lines themselves.

Acknowledgment This work was supported by both the National Scholarship Programme of the Slovak Republic (E.B.) and the Slovak VEGA Grant Agency, Project # 2/0003/16 (M.S.). We are extremely grateful to Zsolt Szab´o for an electronic version of Figure 2.

References [1] Bartnicka, E., Matra´s, A.: 2016, Free Cyclic Submodules in the Context of the Projective Line, Results. Math., Vol. 70, pp. 567–580. [2] Bartnicka, E., Matra´s, A.: 2017, The distant graph of the projective line over a ﬁnite ring with unity, Results. Math., Vol. 72, pp.1943–1958 [3] Hartshorne, R.: 1967, Foundations of Projective Geometry, Benjamin Press. [4] Havlicek, H.: 2012, Divisible Designs, Laguerre Geometry, and Beyond, J. Math. Sci., New York, 186, 882-926. [5] Havlicek, H., and Saniga, M.: 2008, Projective Ring Line of an Arbitrary Single Qudit, Journal of Physics A: Mathematical and Theoretical, Vol. 41, No. 1, 015302 (12pp). [6] Havlicek, H., and Saniga, M.: 2009, Vectors, Cyclic Submodules and Projective Spaces Linked with Ternions, Journal of Geometry, Vol. 92, Nos. 1-2, pp. 79–90. [7] Herzer, A.: 1995, Chain Geometries, In Buekenhout, F., editor, Handbook of Incidence Geometry, pp. 781–842, Elsevier, Amsterdam. [8] Saniga, M., Havlicek, H., Planat, M., and Pracna, P.: 2008, Twin ‘FanoSnowﬂakes’ over the Smallest Ring of Ternions, Symmetry, Integrability and Geometry: Methods and Applications, Vol. 4, Paper 050, 7 pages. [9] Saniga, M., Planat, M., and Pracna, P.: 2008, Projective Ring Line Encompassing Two-Qubits, Theoretical and Mathematical Physics, Vol. 155, No. 3, pp. 905–913.

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[10] Saniga, M., and Pracna, P.: 2010, Space versus Time: Unimodular versus Non-Unimodular Projective Ring Geometries, Journal of Cosmology, Vol. 4, pp. 719–735. Edyta Bartnicka University of Warmia and Mazury Faculty of Mathematics and Computer Science Sloneczna 54 Street, P-10710 Olsztyn Poland E-mail: [email protected] Metod Saniga Astronomical Institute Slovak Academy of Sciences SK-05960 Tatransk´a Lomnica Slovak Republic E-mail: [email protected]

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