We prove that there exist S-spaces containing an arbitrary number of non-isomorphic affine planes of any order. The proof is obtained by constructing some new S-spaces in two different ways. In one case we obtain S-spaces of finite order containing an infinite number of points, while in the other case we obtain S-spaces of infinite order

Sonnino, Angelo

English

Contributions to Discrete Mathematics (E-Journal, University of Calgary)

Volume 3, Number 1, Pages 58–62ISSN 1715-0868S-SPACES FROM FREE EXTENSIONSANGELO SONNINODedicated to the centenary of the birth of Ferenc Ka´rteszi (1907–1989).Abstract. We prove that there exist S-spaces containing an arbitrarynumber of non-isomorphic affine planes of any admissible order. Theproof is obtained by constructing some new S-spaces in two differentways. In one case we obtain S-spaces of finite order containing an infinitenumber of points, while in the other case we obtain S-spaces of infiniteorder.1. IntroductionThe study of S-spaces begun in the early 60’s when E. Sperner [10] in-troduced certain incidence structures similar to ordinary affine spaces, butwith some weaker properties regarding the classical Desargues theorem andthe concept of dimension. Some fairly recent results on S-spaces are in[4, 7, 8, 9], while a good account on the basic properties of S–spaces can befound in [2].A generalised affine space (briefly, an S-space) is an incidence structure Sof “points” and “lines”, together with a binary relation between lines whichis called “parallelism”, satisfying the following axioms:(1) Any two points are incident with exactly one line;(2) All the lines are incident with the same number of points;(3) The parallelism is an equivalence relation;(4) Given a line ` and a point x, there exists exactly one line `′ in Swhich is incident with x and parallel to `.Using Axioms (3) and (4) we find that if two lines `1 and `2 are parallel,then either `1 = `2 or `1 ∩ `2 = ∅.Ordinary affine spaces provide the first examples of S-spaces, while anS-space S which is not an ordinary affine space is called a “proper” S-space.Further, if the number of points of Axiom (2) is finite, say n, then S iscalled a finite S-space of order n.It is well known that the only subspaces of dimension 2 contained in anordinary affine space are Desarguesian affine planes, while this is not true2000 Mathematics Subject Classification. Primary 51A15. Secondary 51E15.Key words and phrases. S-space, free extension, affine space, incidence.c©2008 University of Calgary58S-SPACES FROM FREE EXTENSIONS 59in general when a proper S-space is considered. For a proper S-space S, thefollowing questions arise:• How many non-isomorphic affine planes are contained in S as sub-spaces?• What are the maximum and the minimum number of non-isomorphicaffine planes through a point?This problem was originally posed in a more general setting by Barlotti[2] who defined the regularity parameters of an S-space S, that is, the min-imum number mr and the maximum number Mr of ordinary affine spacesof dimension r through a point of S.In this paper we construct finite S-spaces containing k non-isomorphicaffine planes of given order n for any k < δ +1, with δ denoting the numberof isomorphism classes of affine planes of order n, and show that for such anS-space m2 ≥ k holds with an arbitrarily large number of non-isomorphicaffine planes through each point.2. PreliminariesFrom Axiom (4) it follows that through every point of a finite S-space Sof order n there pass the same number of lines. Let b(x) be the number oflines through each point x of an S-space S. The “dimension” of S is givenby one of the following:• if b(x) = ∞ for any x ∈ S, then S has infinite dimension;• if there is a positive integer r such thatb(x) =nr − 1n− 1for a fixed n ∈ N and any x ∈ S, then S has regular dimension r;• if none of the above cases occurs, then S has no regular dimension.S-spaces with no regular dimension actually exist, see [5, 6], while S-spacesof regular dimension 2 are always ordinary affine planes, see [1, Theorem1.2.1] for instance.In the remainder of this section we recall an inductive method for con-structing S-spaces due to A. Barlotti [2].Let S = (P,L) be a near linear space with set of points P and set oflines L (see [3]), such that no line contains more than s points for a certainpositive integer s. Set{Aj = (Pj , Lj) | j = 1, 2, 3, . . . },where Aj is an incidence structure of “points” and “lines” with point set Pjand line set Lj defined as follows:(1) A0 = S;(2) Ah+1 is obtained from Ah as follows:(a) let F be a family of subsets of Ph such that:(i) no such subset contains two points on a line of Lh;60 ANGELO SONNINO(ii) every two points of Ph belong to exactly one subset of F ;(iii) every subset of F contains k points, with 1 < k ≤ s.If Ah is not an S-space in its own right, then there exists a set Fas above: One example is provided by the set of pairs of pointswhich are not joined by a line of Lh. Once we found such a setF , we consider its subsets as new “lines” that will be added tothose of Lh to obtain an incidence structure A(1)h= (P(1)h, L(1)h),with L(1)h = Lh ∪ F . Then we extend in a natural way theexisting parallelism to these new lines by considering each ofthem parallel to itself. Doing so, we introduce some new classesof parallelism, each consisting of a single line.(b) Add to any line of L(1)hcontaining k < s points s−k new points.This yields a new incidence structure A(2)h = (P(2)h , L(2)h ).(c) Choose some subsets of P(2)h such that no two points in each ofthem are on a line or on two parallel lines of L(2)h . Contract eachof these subsets to one point, in order to obtain a new incidencestructure A(3)h = (P(3)h , L(3)h ).(d) Let `1 and `2 be two lines such that no line parallel to one ofthem meets a line parallel to the other. If such pairs of linesexist, then define a new parallelism class containing `1, `2, allthe lines parallel to `1 and all the lines parallel to `2. This yieldsnew incidence structure A(4)h = (P(4)h , L(4)h ).(e) Let ` ∈ L(4)h . For every point x ∈ P(4)h not contained in a lineparallel to ` add a new line `x (initially containing the point xonly) to the parallelism class of `. The incidence structure soconstructed is Ah+1.An incidence structure At, t ∈ N, obtained from a near linear space S asabove is called an extension of order t of S. Such an extension is called afree extesion if every subset of 2a contains exactly two points and neitherthe contraction of 2c, nor the modification of 2d are performed.Theorem 2.1 (Barlotti). The incidence structureS = limh→∞Ahis an S-space.Using free extensions, Barlotti was able to construct a class of S-spaceswith regularity parameter M2 = 0, that is, S-spaces containing no affineplanes.3. The required S-spacesLet δ denote the number of non-isomorphic affine planes of a certain ordern.S-SPACES FROM FREE EXTENSIONS 61Theorem 3.1. For every positive integer k < δ + 1, let {pi0, pi1, . . . , pik−1}be a set of non-isomorphic affine planes of order n. Then there exists anS-space S of order n containing all the pij as subspaces. Furthermore, Shas regularity parameter m2 ≥ k.Proof. For a prime power n, let A0 = (P,L) be a near linear space whoselongest line contains at most k ≤ n points, but containing some lines ofsize less than n. For h > 0 let Ah be a free extension of A0, and A(2)h=(P(2)h , L(2)h ) the incidence structure obtained after performing 2b on Ah. Letj be an integer with 1 ≤ j ≤ k. If h ≡ j (mod k), then for each pointx ∈ P(2)hnot contained in any affine plane isomorphic to pij add a set B ofn2−1 new points in such a way that {x}∪B yields an affine plane isomorphicto pij. Denote the resulting incidence structure by Ah+1.After m such extensions of Ah, with 1 < m < k, we end up with anincidence structure Ah+m containing points which are in no affine planeisomorphic to pij ; however, these points can be included in such affine planesextending Ah+m again. The incidence structureN = limh→∞Ahis a finite S-space of order q satisfying all the required conditions. Theexistence of the parallelism is granted by the fact that a free extesion includes2e. Note that the S-space N arising from Theorem 3.1 is a finite S-spaceof order n containing an infinite number of points. Now we are going toconstruct S-spaces of infinite order instead, in order to prove the followingresult.Theorem 3.2. There exist S-spaces satisfying M2 = m2 = ∞.Proof. As in the proof of Theorem 3.1, we start off with a near linear spaceA0 whose lines have length at most s. For every h ≥ 0, obtain a freeextension of Ah by adding s + h − r points to each line of L(1)h containingr ≤ s points, and denote by Bh+1 = (P′h+1, L′h+1) the resulting incidencestructure. If h is an integer such that no affine plane of order s + h exists,then put Bh+1 = Ah+1 and go on; otherwise, for every point x ∈ P′h+1 add(s + h)2 − 1 more points in such a way that these points together with xconstitute an affine plane of order s + h. The resulting S-spaceM = limh→∞Ahhas infinite order, and contains finite affine planes of any admissible order.Further, the condition M2 = m2 = ∞ is an obvious consequence of theconstruction. We remark that all the S-spaces arising from both Theorems 3.1 and 3.2have infinite dimension.62 ANGELO SONNINOReferences1. A. Barlotti, Some topics in finite geometrical structures, Institute of Statistics MimeoSeries, no. 439, University of North Carolina.2. , Alcuni risultati nello studio degli spazi affini generalizzati di Sperner, Rend.Sem. Mat. Univ. Padova 35 (1965), 18–46.3. L. M. Batten, Combinatorics of Finite Geometries, Cambridge University Press, Cam-bridge, 1986.4. A. Blunck, A new approach to derivation, Forum Math. 14 (2002), no. 6, 831–845.5. R. C. Bose, On the application of finite projective geometry for deriving a certain seriesof balanced Kirkman arrangements, Calcutta Math. Soc. Golden Jubilee Commemo-ration Vol. (1958/59), Part II, Calcutta Math. Soc., Calcutta, 1963, pp. 341–354.6. P. Quattrocchi, Un metodo per la costruzione di spazi affini generalizzati di Sperner,Matematiche (Catania) 22 (1967), 1–9.7. A. Sonnino, A new class of Sperner spaces, Pure Math. Appl. 9 (1998), no. 3-4, 451–462.8. , Cryptosystems based on latin rectangles and generalised affine spaces, Rad.Mat. 9 (1999), no. 2, 177–186.9. , Two methods for constructing S-spaces, Atti Sem. Mat. Fis. Univ. Modena51 (2003), no. 1, 65–71.10. E. Sperner, Affine Ra¨ume mit schwacher Inzidenz und zugeho¨rige algebraische Struk-turen, J. Reine Angew. Math. 204 (1960), 205–215.Dipartimento di Matematica e Informatica, Universita` della Basilicata,Viale dell’Ateneo Lucano 10, 85100 Potenza, ItaliaE-mail address: angelo.sonnino@unibas.it

S-spaces from free extensions

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S-spaces from free extensions

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Contributions to Discrete Mathematics (E-Journal, University of Calgary)