ON SOME HYPERBOLIC PLANES FROM FINITE PROJECTIVE PLANES

LetΠ= (P,L,I) be a finite projective plane of ordern, and letΠ′ = (P ′,L′, I′) be a subplane of Π with orderm which is not a Baer subplane (i.e., n≥m2+m). Consider the substructure Π0 = (P0,L0, I0) with P0 = P\{X ∈ P |XIl, l∈ L′}, L0 = L\L′, where I0 stands for the restriction of I to P0×L0. It is shown that everyΠ0 is a hyperbolic plane, in the sense of Graves, if n≥m2+m+1+ √ m2+m+2. Also we give some combinatorial properties of the line classes in Π0 hyperbolic planes, and some relations between its points and lines. 2000 Mathematics Subject Classification. Primary 51E20; Secondary 05B25.


Introduction.
In this paper, points are denoted by capital letters (usually P,Q), lines are denoted by lower-case letters (usually l), sets ᏼ and ᏸ denote the sets of points and lines, respectively, Ᏽ denotes the incidence relation on points and lines (therefore Ᏽ ⊂ ᏼ × ᏸ).The triple (ᏼ, ᏸ, Ᏽ) is called a geometric structure, if ᏼ ∩ ᏸ =Φ.If (P , l) ∈ Ᏽ then P is on l or l passes through P and it is denoted by P ∈ l or P Ᏽl.Similarly if (P , l) ∉ Ᏽ then P is not on l and it is denoted by P ∉ l.If ᏼ and ᏸ are finite sets, the geometric structure (ᏼ, ᏸ, Ᏽ) is called finite.
It is well known that there are alternative systems of axioms for hyperbolic spaces.For instance, Graves [3] introduced the following definition (see [1,2,5,6]).
A finite hyperbolic plane is a finite geometric structure (ᏼ, ᏸ, Ᏽ) such that (G1) Two distinct points lie on one and only one line.(G2) There are at least two points on each line.(G3) Through each point X not on a line l there pass at least two lines not meeting (parallel to) l.
(G4) There exist at least four points, no three of which are collinear.(G5) If a subset of ᏼ contains three non-collinear points and all the lines through any pair of its points, then this subset contains all points of ᏼ.
In this paper, we construct a class of hyperbolic planes using the non-Baer subplanes of the projective planes of finite order.Thus, in a sense, we find a connection between the non-Baer subplanes of finite projective plane and some hyperbolic planes from that plane by certain deletion.
2. Construction of finite hyperbolic spaces Π 0 .Let Π = (ᏼ, ᏸ, Ᏽ) be a finite projective plane of order n with a non-Baer subplane Π = (ᏼ , ᏸ , Ᏽ ) of order m.Then it is well known that n ≥ m 2 + m . (2.1) obtained by removing all lines of Π 0 with incidence points.Thus, The following theorem is an immediate consequence of the construction of Π 0 .A line which contains exactly one point of Π 0 is said to be a tangent line and a line which contains no points of Π 0 is called an exterior line.Proof.Let l 0 ∈ ᏸ 0 and l denotes the extended line of l 0 in Π.Then l is either a tangent or an exterior line.If l is an exterior line, then l has m 2 +m+1 deleted points.Thus l 0 has n − m 2 − m points.Otherwise l must be a tangent line and therefore it has m 2 + 1 deleted points.Thus, if l is a tangent line, then it has n − m 2 points.
It is trivial from Theorem 2.1(i) that, in Π 0 , (G1) is satisfied.Any line of Π 0 contains at least n−m 2 −m points, by Theorem 2.1(iv).By (G2), it must be greater than 2, that is, (2.2) Notice that (2.2) is stronger than (2.1).Hence, any line l of Π 0 has at least m 2 + 1 deleted points, in Π 0 there are at least m 2 + 1 parallel lines through any point X, X ∈ l.Since m ≥ 2, through each point X not on a line l there pass at least five lines parallel to l.Hence, Π 0 satisfies properties (G1), (G2), and (G3), if (2.2) holds.That existence of four points no three of which are collinear is obvious from the definition of Π 0 .
Finally, we investigate when the last axiom is satisfied in Π 0 .Let ⊂ ᏼ 0 contain three non-collinear points A, B, C. We consider the lines AB, AC, and BC.Then contains all of the points on the lines AB, AC, and BC, and all points on the lines through pairs of distinct points of .Each of the lines has at least n − m 2 − m points in .Thus, there are at least n − m 2 − m lines in Π 0 through A and meeting the line BC.contains at least, (n − m 2 − m)(n − m 2 − m − 1) + 1 points, since each of the above lines contains at least n − m 2 − m − 1 points other than the point A. Now, let X be a point of Π 0 not on a line that joins the point A to a point of BC.X is in if there exists a line which contains X and at least two of the above since X is on exactly n+1 lines and these lines contains all points of Π 0 .This inequality is valid when But, equations (2.2) and (2.3) cannot be true at the same time.Therefore (2.3) is eliminated.Thus the following theorem is obtained.
Theorem 2.3.Let Π = (ᏼ, ᏸ, Ᏽ) be a finite projective plane of order n with a non-Baer subplane Π = (ᏼ , ᏸ , Ᏽ ) of order m.Then the substructure Π 0 = (ᏼ 0 , ᏸ 0 , Ᏽ 0 ), is a hyperbolic plane, in the sense of Graves, if (2.6) 3. Some properties of Π 0 .The following theorem is an immediate consequence of the construction of Π 0 .We define the following line classes; which consist of tangent and exterior lines of Π 0 , respectively.We call C t as tangent lines class, C e as exterior lines class.In the next section, we give some combinatorial properties of the line classes in Π 0 by using the technique of [4].

Parallel line classes of Π 0 hyperbolic planes.
A class of the lines every two of which are parallel is called parallel line class.All lines of Π 0 passing through any deleted point P of Π form a parallel line class.This parallel line class is called parallel class determined by P or parallel class of type (P ).A line together with all lines passing through a deleted point Q which is not on l and cutting l in the deleted points in Π form a parallel line class.But many parallel classes can be found containing this parallel line class.The intersection of all parallel classes containing the mentioned class of lines is called parallel class determined by l and Q, or parallel class of type (l, Q).
As, there might be other parallel classes apart from the above ones, it is convenient to call the parallel classes of type (P ) and (l, Q) as obvious parallel classes.Proof.The necessary and sufficient condition for a point P to be a deleted point is that either P ∈ ᏼ or P ∉ ᏼ , P Ᏽl ∈ ᏸ .Therefore, (i) if P ∈ ᏼ , then the number of lines of ᏸ 0 passing through P is n − m.As all of these n − m lines pass through the deleted point P , |(P )| = n − m.
(ii) If P ∉ ᏼ , P Ᏽl ∈ ᏸ , then the number of lines of Π 0 passing through a deleted external point is the required number.n + 1 lines pass through P except one, none of these lines do not belong to Π .Therefore, the number of lines of Π 0 passing through P in Π is n.Theorem 4.2.We denote the minimum number of lines belonging to the parallel class of (l, P ) type by min |(l, P )|.Then, Proof.Let l be any line of if P ∈ ᏼ , the number of deleted points on l is m 2 + 1.Furthermore, m + 1 lines pass through P in Π and these lines are the deleted lines.Therefore, together with the line l at least m 2 + 1 − (m + 1) + 1 = m 2 − m + 1 lines belong to (l, P ) type.
(b) If P ∉ ᏼ , the number of deleted points on l is m 2 + 1.If we join this m 2 + 1 points with P not incident on l, then the obtained m 2 +1 lines meet l at deleted points in Π.Since one of these lines is a deleted line, there are at least m 2 +1 (l, P )-type lines.
(ii) If l ∈ C d , then (a) if P ∉ ᏼ , then m 2 +m+1 points are deleted from l. Join these points to P ∉ l.The m 2 + m + 1 lines which are obtained by joining P ∉ l to deleted points from l meets l on deleted points in Π.Since one of these lines is a deleted line, together with l at least m 2 + m + 1 lines are in type (l, P ).
(b) If P ∈ ᏼ , then there are m+1 lines passing through P in Π .For this, there are m + 1 deleted lines among the m 2 + m + 1 lines obtained by joining P to the deleted m 2 + m + 1 points from l. Hence, together with l there are at least m 2 + 1 lines in type (l, P ).Proof.Since l ∈ C t , the intersection of l and Π has one and only one point.Since m+1 lines pass through this point, the remaining m 2 lines meet l on different points.Since every deleted point corresponds to a parallel line class of type (P ) and l has m 2 + 1 deleted points, line l 0 belongs to m + 1(P )-type parallel class.
On the other hand, the number of deleted points which is not on l is, Theorem 4.5.Let ᏸ 0 denotes the set of lines of Π 0 , i be any parallel line class type and f i (l), l ∈ ᏸ 0 be the number of parallel line classes type i.Then the following equations are valid: Proof.Any line l ∈ ᏸ 0 belongs to a parallel line class of type (P ), as much as the number of deleted points from l and type (l, P ) as much as the number of deleted points from Π which is not on l.Therefore, there are parallel line classes of type (P ) or type (l, P ) as much as the number of deleted points from Π. Since the number of deleted points from Π is (m 2 + m + 1)(n + 1 − m), we obtain The sum l∈ᏸ f (P ) (l) is the number of total flags which are obtained from deleted points and the lines of Π 0 passing through these deleted points.This sum can be written as follows:  Total anti-flag numbers of deleted points and lines of Π 0 not passing through these points is l∈ᏸ 0 f (l,P ) (l).Hence,

Isomorphism.
Let Π be a projective plane of order n and Π , Π be subplanes of Π with order m, and n ≥ m 2 + m + 1 + √ m 2 + m + 2. Then according to Theorem 2.1, we can construct the hyperbolic planes Π 0 , Π 0 by deleting, respectively, the lines of Π , Π together with incident points.Then we can give the following obvious consequence.

Theorem 2 . 1 .
The following properties are valid: (i) Two distinct points of Π 0 lie on one and only one line of Π 0 .(ii) There are exactly n 2 + n − m 2 − m lines in Π 0 .(iii) There are exactly (n − m)(n − m 2 ) points in Π 0 .(iv) At least n − m 2 − m points lie on any line of Π 0 .

Theorem 2 . 2 .
Any line of Π 0 contains exactly either n − m 2 − m points or n − m 2 points.

Theorem 3 . 2 .
The line of Π 0 which is contained in C t or C e contains at most n − 4 or n − 6 points, respectively.Proof.It is clear, if the reality of m ≥ 2 is used with the definition of C t and C e .

Theorem 4 . 1 .
There are n or n − m lines in a parallel line class of type (P ) of Π 0 .

. 2 )Theorem 4 . 4 .
Therefore line l belongs to m(m + 1)(n − m) + n(l, P )-type parallel class.Now we give a theorem which can be proved like the previous theorem.The line l ∈ C d of Π 0 belongs to m 2 + m + 1(P )-type and (m 2 + m + 1)(n − m)(l, P )-type parallel classes.