Metod Saniga- Projective Ring Lines and Finite Generalized Quadrangles

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    PROJECTIVE RING LINES ANDFINITE GENERALIZED

    QUADRANGLES(FPGQT, Tatranska Lomnica, August 2, 2007)

    METOD SANIGA

    Astronomical Institute of the Slovak Academy of SciencesSK-05960 Tatranska Lomnica, Slovak Republic

    ([email protected])

    AN OVERVIEW OF THE TALK

    1. Introduction2. Rudiments of Ring Theory

    Denition of a(n Associative) Ring

    Units, Zero-Divisors, Characteristic, Fields Ideals, Jacobson radical, Quotient Rings

    Mappings: Ring Homo- and Isomorphism

    Examples of (Finite) Rings:Abstract and Concrete/Illustrative

    3. Projective Line over a Ring GL(2, R) and Pair Admissibility

    Projective Line over a RingR, P R (1)

    Neighbour/Distant Relations

    The Fine Structure of P R (1): Points of Type I and II

    Illustrative Examples of Finite Projective Ring Lines Classication of Projective Ring Lines up to Order 63

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    4. Finite Generalized Quadrangles

    Denition, Order, Grids and Dual Grids

    Some Distinguished Subsets: Perp-sets, Ovoids, Triads, etc. Geometric Hyperplanes and Associated Veldkamp Spaces

    Smallest Generalized Quadrangle GQ(2,2) and Its Veldkamp Space

    GQ(2,2) and the Projective Line over the Full Two-by-Two Matrix Ringwith Entries in GF(2)

    GQ(2,2) and Icosahedron

    5. References

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    1. Introduction

    Finite projective geometries, ring lines and generalized quadrangles inparticular, represent a well-studied, important and venerable branch of algebraic geometry.

    Although these geometries are endowed with a number of fascinatingproperties having no analogues in the classics, it may well come asa surprise that they have so far successfully evaded the attention of physicists and scholars of other natural sciences.

    The purpose of the talk is to reveal the beauty of these objects.

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    2. Rudiments of Ring Theory

    Denition of a(n Associative) Ring

    A ring is a set R (or, more specically, (R, + ,)) with two binary op-erations, usually called addition (+) and multiplication (), such that Ris an abelian group under addition and a semi group under multiplication,

    with multiplication beingboth left and right distributive over addition. (Itis customary to denote multiplication in a ring simply by juxtaposition,using ab in place of a b.)

    A ring in which the multiplication is commutative is acommutative ring.

    A ring R with a multiplicative identity 1 such that 1r = r 1 = r for allr R is a ring with unity .

    A ring containing a nite number of elements is anite ring; the num-ber of its elements is called itsorder .

    In what follows the word ring will always mean a commutative ring with unity

    unless stated otherwise.

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    Units, Zero-Divisors, Characteristic, Fields

    An element r of the ring R is a unit (or an invertible element) if thereexists an elementr 1 such that rr 1 = r 1r = 1. This element, uniquelydetermined by r , is called the multiplicative inverse of r . The set of unitsforms a group under multiplication.

    A (non-zero) elementr of R is said to be a (non-trivial)zero-divisor if there exists s = 0 such that sr = rs = 0; 0 itself is regarded as trivialzero-divisor.

    An element of a nite ring is either a unit or a zero-divisor. A unitcannot be a zero-divisor.

    A ring in which every non-zero element is a unit is aeld ; nite (or Galois)elds, often denoted by GF(q), have q elements and exist only forq = pn ,where p is a prime number andn a positive integer.

    The smallest positive integers such that s1 = 0, where s1 stands for1 + 1 + 1 + . . . + 1 (s times), is called thecharacteristic of R; if s1 is neverzero, R is said to be of characteristic zero.

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    Ideals, Jacobson radical, Quotient Rings

    An ideal I of R is a subgroup of (R, +) such that a I = I a I for alla R. Obviously,{0} and R are trivial ideals; in what follows the wordideal will always mean proper ideal, i.e. an ideal different from either of the two. A unit of R does not belong to any ideal of R; hence, an idealfeatures solely zero-divisors.

    An ideal of the ringR which is not contained in any other ideal butRitself is called amaximal ideal.

    If an ideal is of the formRa for some elementa of R it is called aprincipal ideal, usually denoted by a .

    A very important ideal of a ring is that represented by the intersectionof all maximal ideals; this ideal is called theJacobson radical .

    A ring with aunique maximal ideal is alocal ring.

    Let R be a ring and I one of its ideals. ThenR R/ I = {a + I | a R}together with addition (a + I ) + ( b + I ) = a + b + I and multiplication(a + I )(b+ I ) = ab + I is a ring, called thequotient , or factor , ring of Rwith respect to I ; if I is maximal , then R is a eld .

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    Mappings: Ring Homo- and Isomorphism

    A mapping : R S between two rings (R, + ,) and (S,,) is a ringhomomorphism if it meets the following constraints:

    (a + b) = (a) (b),(a b) = (a)(b) and(1) = 1 for any two elementsa and b of R.

    From this denition it follows that(0) = 0, ( a) = (a), a unit of Ris sent into a unit of S and the set of elements{a R | (a) = 0}, calledthe kernel of , is an ideal of R.

    A canonical , or natural , map : R R R/ I dened by (r ) = r + I is clearly a ring homomorphism with kernelI .

    A bijective (i.e., one-to-one and onto) ring homomorphism is called a ringisomorphism; two ringsR and S are called isomorphic, denoted byR = S ,if there exists a ring isomorphism between them.

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    Examples of (Finite) Commutative Rings:Abstract

    A polynomial ring,R[x], viz. the set of all polynomials in one variablexand with coefficients in a ringR.

    The ring R that is a (nite) direct product of rings,R R1R2. . .Rn , where both addition and multiplication are carried out componentwiseand where the individual rings need not be the same.

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    Examples of (Finite) Commutative Rings:Concrete/Illustrative

    GF (2): Order 2, Characteristic 2, a eld 0 10 0 11 1 0

    0 10 0 01 0 1

    Note that 1 + 1 = 0 implies +1 = 1, which is valid for any ring of characteristic two.

    GF (4 = 22) = GF (2)[x]/ x2 + x + 1 : Order 4, Characteristic 2, a eld

    0 1 x x + 10 0 1 x x + 11 1 0 x + 1 xx x x + 1 0 1

    x + 1 x + 1 x 1 0

    0 1 x x + 10 0 0 0 01 0 1 x x + 1x 0 x x + 1 1

    x + 1 0 x + 1 1 x

    GF (2)[x]/ x2 : Order 4, Characteristic 2, not a eld

    0 1 x x + 10 0 1 x x + 11 1 0 x + 1 xx x x + 1 0 1

    x + 1 x + 1 x 1 0

    0 1 x x + 10 0 0 0 01 0 1 x x + 1x 0 x 0 x

    x + 1 0 x + 1 x 1

    A unique maximal (and also principal) ideal:I x = {0, x} its a localring. Both GF (4 = 22) and GF (2)[x]/ x2 have the same addition table.

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    Z 4: Order 4, Characteristic 4, not a eld

    0 1 2 30 0 1 2 31 1 2 3 02 2 3 0 13 3 0 1 2

    0 1 2 30 0 0 0 01 0 1 2 32 0 2 0 23 0 3 2 1

    A unique maximal (and also principal) ideal:I x = {0, 2} its a localring. Both Z 4 and GF (2)[x]/ x2 have the same multiplication table.

    GF (2)[x]/ x(x + 1) = GF (2)GF (2):Order 4, Characteristic 2, not a eld

    0 1 x x + 10 0 1 x x + 11 1 0 x + 1 xx x x + 1 0 1

    x + 1 x + 1 x 1 0

    0 1 x x + 10 0 0 0 01 0 1 x x + 1x 0 x x 0

    x + 1 0 x + 1 0 x + 1

    Two maximal (and principal as well) ideals:I x = {0, x} and I x+1 ={0, x + 1} it is not a local ring. Each element except 1 is a zero-divisor.Has the same addition table as both GF (4) and GF (2)[x]/ x2 .

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    R Z 4Z 4: Order 16, Characteristic 4, not a eld a b c d e f g h i j k l m n p qa a b c d e f g h i j k l m n p q

    b b c d a h k n i j e l m f p q gc c d a b i l p j e h m f k q g nd d a b c j m q e h i f k l g n pe e h i j f g a k l m n p q b c df f k l m g a e n p q b c d h i jg g n p q a e f b c d h i j k l mh h i j e k n b l m f p q g c d ai i j e h l p c m f k q g n d a b

    j j e h i m q d f k l g n p a b ck k l m f n b h p q g c d a i j el l m f k p c i q g n d a b j e h

    m m f k l q d j g n p a b c e h in n p q g b h k c d a i j e l m f p p q g n c i l d a b j e h m f kq q g n p d j m a b c e h i f k l

    a b c d e f g h i j k l m n p qa a a a a a a a a a a a a a a a ab a b c d a a a b c d b c d b c dc a c a c a a a c a c c a c c a cd a d c b a a a d c b d c b d c be a a a a e f g e e e f f f g g gf a a a a f a f f f f a a a f f f g a a a a g f e g g g f f f e e eh a b c d e f g h i j k l m n p qi a c a c e f g i e i l f l p g p

    j a d c b e f g j i h m l k q p nk a b c d f a f k l m b c d k l ml a c a c f a f l f l c a c l f l

    m a d c b f a f m l k d c b m l kn a b c d g f e n p q k l m h i j p a c a c g f e p g p l f l i e iq a d c b g f e q p n m l k j i h

    .

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    From these tables it follows thata and h are, respectively, the addition andmultiplication identities (0 and 1) of the ring,

    R = {h 1,j ,n,q } (1)

    is the set of units and

    R \ R = {a 0,b,c,d,e,f ,g, i ,k, l ,m,p } (2)

    that of zero-divisors. The latter comprises two maximal ideals,

    I 1 = {a,c,f , l ,b,d,k,m }, (3)

    I 2 = {a,c,f , l ,e ,g, i ,p }, (4)

    yielding anon -trivial Jacobson radical

    J = I 1 I 2 = {a,c,f , l }, (5)

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    Full Matrix Ring M 2(GF (2))) and Its Subrings(Order 16, Characteristic 2,Non-commutative)

    The full two-by-two matrix ring withGF (2)-valued coefficients,

    R = M 2(GF (2)) | ,, , GF (2) . (6)

    In an explicit form:

    UNITS: Invertible matrices (i. e., matrices with non-zero determinant). Theyare of two distinct kinds: those which square to 1,

    1 1 00 1 , 2 0 11 0 , 9

    1 10 1 , 11

    1 01 1 , (7)

    and those which square to each other,

    12 0 11 1 , 13 1 1

    1 0 . (8)

    ZERO-DIVISORS: Matrices with vanishing determinant. These are also of two different types:nil potent, i. e. those which square to zero,

    3 1 11 1 , 8 0 10 0 , 10

    0 01 0 , 0

    0 00 0 , (9)

    and idem potent, i. e. those which square to themselves,

    4 0 01 1 , 5 1 0

    1 0 , 6 0 1

    0 1 , 7 1 1

    0 0 , (10)

    14 0 00 1 , 15 1 00 0 . (11)

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    The structure of this full matrix ring can be well understood from theaccompanying colour gure featuring its most important subrings, namelythose isomorphic to

    GF (4) (yellow), GF (2)[x]/ x2 (red),

    GF (2)GF (2) (pink) and to

    the non-commutative ring of 8/6 type (green).

    Irrespectively of colour, the dashed/dotted lines join elements represented

    by upper/lower triangular matrices, while the solid lines link elements rep-resented by diagonal parity preserving matrices.

    It is worth mentioning a very interesting symmetry of the picture. Namely,the dpp ring of 8/6 type (solid green) incorporates both the upper andlower triangular matrix rings isomorphic toGF (2)GF (2), while, in turn,the dpp GF (2)GF (2) ring (solid pink) is the intersection of the upperand lower triangular matrix rings of 8/6 type.

    It is also to be noted that GF (4) has only one representative, the dppset, whereas each of the remaining types havethree distinct (namely up-per and lower triangular, and dpp) representatives. The shaded circlesdenote non-trivial idempotents.

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    0

    57

    1514

    46

    1038

    1312

    1129

    1

    M (GF(2))2

    zd

    u

    Figure 1: The subrings of M 2 (GF(2)).

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    3. Projective Line over a Ring

    GL( 2, R) and Pair Admissibility

    Given a ring R with unity and GL(2, R), the general linear group of

    invertible two-by-two matrices with entries inR,

    a pair (, ) R2 is called admissible over R if there exist , Rsuch that

    GL(2, R). (12)

    or, equivalently,

    det R. (13)

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    Projective Line over a Ring R , P R (1)The projective line overR , P R (1):the set of classes of ordered pairs ( , ),where is a unit and (, ) is admissible .

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    Neighbour/Distant Relations

    Such a line carries two non-trivial, mutually complementary relations of neighbour and distant.In particular, its two distinct points X :=( , ) and Y :=( , ) arecalled neighbour (or, parallel ) if

    /GL(2, R) det

    R\ R

    (14)

    and distant otherwise, i. e., if GL(2, R) det

    R

    . (15)

    The neighbour relation is reexive (every point is obviously neighbour to itself) and symmetric (i.e., if X is neighbour toY then Y is neighbour toX

    too), but, in general, not transitive (i.e., X being neighbour toY and Y being neighbour

    to Z does not necessarily mean thatX is neighbour toZ ).

    Given a point of P R (1), the set of all neighbour points to it will be calledits neighbourhood .

    Obviously, if RL is a eld then neighbour simply reduces to identical(and, hence, distant to different); for Eq. (8) reduces to

    det = = 0 (16)

    which indeed implies

    = and = . (17)

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    The Fine Structure of P R (1): Points of Type I and II

    P R (1) comprises, in general, two distinct groups of points.

    Points of Type I : the points represented by coordinates whereat least one entry is a unit .

    It is easy to verify that for any nite commutative ring this number isalways equal to the sum of the total number of elements of the ring andthe number of its zero-divisors; for, indeed, if is a unit then we can always select in such a way

    that ( , ) (1, ), where R and if only is a unit then ( , ) ( , 1), where R\ R.

    Points of Type II : the points represented by coordinates whereboth entries are zero-divisors .

    These points exist only if the ring hastwo or more maximal ideals andtheir number depends on the properties and interconnection between these

    ideals. For ( , ), with , being both zero-divisors of R, to representa point of P R (1), Eq. (7) requires

    det = R. (18)

    This constraint cannot be met if R contains just a single maximal idealI ,because I implies I , I implies I , which implies thatthe whole expression I and, so, isnot a unit.

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    Illustrative Examples of Finite Projective Ring Lines

    R = GF (q):the line contains q (total # of elements) + 1 (# of zero-divisors) points,any two of them being distant.

    R = Z 4:the line contains 4+2 = 6 points, forming three pairs of neighbours, namely(page 9):

    (1, 0) and (1, 2)(0, 1) and (2, 1)(1, 1) and (1, 3)

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    R = GF (2)[x]/ x(x + 1) = GF (2)GF (2):the line is endowed with nine points (page 9), out of which there are

    seven of the rst kind,

    (1, 0), (1, x), (1, x + 1) , (1, 1),(0, 1), (x, 1), (x + 1, 1),

    and two of the second kind,(x, x + 1), (x + 1, x).

    The neighbourhoods of three distinguished pairwise distant pointsU : (1, 0),V : (0, 1) and W : (1, 1) here read

    U : U 1 : (1, x), U 2 : (1, x + 1) , U 3 : (x, x + 1) , U 4 : (x + 1, x), (19)

    V : V 1 : (x, 1), V 2 : (x + 1, 1), V 3 : (x, x + 1) , V 4 : (x + 1, x), (20)

    and

    W : W 1 : (1, x), W 2 : (1, x + 1), W 3 : (x, 1), W 4 : (x + 1, 1). (21)

    From these expressions, and the fact thatGL(2, R) acts transitively ontriples of mutually distant points, we nd that the neighbourhood of any point of this line comprises four distinct

    points, the neighbourhoods of anytwo distant points have two points in

    common (which again implies non-transitivity of the neighbour relation)and

    the neighbourhoods of anythree mutually distant points are disjointas illustrated in the gure; note that in this case there exist no Jacobsonpoints, i. e. the points belonging solely to a single neighbourhood, due tothe trivial character of the Jacobson radical,J = {0}.

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    R = Z 4Z 4: the line contains altogether thirty-six points;twenty-eight of Type I

    (1, 0), (1, b), (1, c), (1, d), (1, e), (1, f ), (1, g), (1, i ), (1, k ), (1, l ), (1, m ), (1, p),(0, 1), (b, 1), (c, 1), (d, 1), (e, 1), (f, 1), (g, 1), (i, 1), (k, 1), (l, 1), (m, 1), ( p, 1),(1, 1), (1, j ), (1, n ), (1, q); (22)

    and eight of Type II(e, b), (e, k ), (i, b), (i, k ), (b, e), (k, e ), (b, i), (k, i ); (23)

    The three distinguished, pairwise distant points of the line, viz.U :=(1, 0), V := (0, 1), W := (1, 1), have the following neighbourhoods

    U : (1, b), (1, c), (1, d), (1, e), (1, f ), (1, g), (1, i ), (1, k ), (1, l ), (1, m ), (1, p),

    (e, b), (e, k ), (i, b), (i, k ), (b, e), (k, e ), (b, i), (k, i ) (24)

    V : (b, 1), (c, 1), (d, 1), (e, 1), (f, 1), (g, 1), (i, 1), (k, 1), (l, 1), (m, 1), ( p, 1),(e, b), (e, k ), (i, b), (i, k ), (b, e), (k, e ), (b, i), (k, i ) (25)

    W : (1, b), (1, d ), (1, e), (1, g), (1, i ), (1, k ), (1, m ), (1, p),(b, 1), (d, 1), (e, 1), (g, 1), (i, 1), (k, 1), (m, 1), ( p, 1),(1, j ), (1, n ), (1, q). (26)

    One thus sees that each neighbourhood features nineteen points and has three Jacobson

    points (underlined), the neighbourhoods pairwise overlap in eight points, have no common element if considered altogether and there exists no point of the line that would be simultaneously distant

    to all the three distinguished points.

    Employing again the fact thatGL(2, R) acts transitively on triples of mu-tually distant points, these properties can be extended toany triple of mutually distant points.

    A nice conic representation of the line exhibiting all these properties is

    given in the following gure, where every bullet representstwo distinctpoints of the line, while each of the three small circles representsthreeJacobson points.

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    R = GF (2)GF (2)GF (2):The line possesses twenty-seven points, twelve of Type II; the neighbour-hood of any point of the line features eighteen distinct points, the neigh-

    bourhoods of any two distant points share twelve points and the neigh-bourhoods of any three mutually distant points have six points in common as depicted in the gure.

    As in the case of the lines dened over GF(2) GF(2) and Z 4 Z 4,the neighbour relation is not transitive; however, a novel feature, not en-countered in the previous cases, is here a non-zero overlapping between theneighbourhoods of three pairwise distant points, which can be attributedto the existence of three maximal ideals of the ring.

    (Every small bullet representstwo distinct points of the line, while the bigbullet at the bottom stands for as many assix different points.)

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    Omitting three distinguished points, the remaining twenty-four points of the line split into two sets of twelve exhibiting intriguing structures in termsof the neighbour/distant relation as illustrated in the gure.

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    P 1(M 2(GF (2)))

    Checking rst for admissibility and then grouping the admissible pairs left-proportional by a unit into equivalence classes (of cardinality six each), wend that P 1(M 2(GF (2))) possesses altogether 35 points, with the followingrepresentatives of each equivalence class:

    (1, 1), (1, 2), (1, 9), (1, 11), (1, 12), (1, 13),(1, 0), (1, 3), (1, 4), (1, 5), (1, 6), (1, 7), (1, 8), (1, 10), (1, 14), (1, 15),(0, 1), (3, 1), (4, 1), (5, 1), (6, 1), (7, 1), (8, 1), (10, 1), (14, 1), (15, 1),(3, 4), (3, 10), (3, 14), (5, 4), (5, 10), (5, 14), (6, 4), (6, 10), (6, 14).(27)

    One can easily recognize that the representatives in the rst row of the lastequation have both entries units (1 being, obviously, unity/multiplicativeidentity), those of the second and third row have one entry unit(y) and theother a zero-divisor, whilst all pairs in the last row feature zero-divisors inboth the entries.

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    Specic Subconguration of P 1(M 2(GF (2))), aka GQ(2,2)

    Let consider two distant points of the line. Taking into account the above-mentioned three-distant-transitivity of GL(2, R), we can take these, with-out any loss of generality, to be the pointsU := (1, 0) and V := (0, 1).Next we pick up all those points of the line which are

    either simultaneously distant or

    simultaneously neighbor

    to U and V . We nd the following six points

    C 1 = (1, 1), C 2 = (1, 2), C 3 = (1, 9),C 4 = (1, 11), C 5 = (1, 12), C 6 = (1, 13), (28)

    to belong to the rst family, whereas the second family comprises the fol-lowing nine points

    C 7 = (3, 4), C 8 = (3, 10), C 9 = (3, 14),C 10 = (5, 4), C 11 = (5, 10), C 12 = (5, 14),C 13 = (6, 4), C 14 = (6, 10), C 15 = (6, 14). (29)

    One readily checks that the points of our special subset of P 1(M 2(GF (2)))are related with each other as shown in Table 2; from this table it canreadily be discerned that

    to every point of the conguration there are six neighbor and eightdistant points, and that

    the maximum number of pairwise neighbor points is three.

    Hence, our subconguration is nothing but thegeneralized quadrangle of order two, GQ(2,2),

    in disguise!

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    Table 1: The distant and neighbor (+ and , respectively) relation between the points of the conguration.The points are arranged in such a way that the last nine of them (i. e., C 7 to C 15 ) form the projective line overGF (2) GF (2).

    C 1 C 2 C 3 C 4 C 5 C 6 C 7 C 8 C 9 C 10 C 11 C 12 C 13 C 14 C 15C 1 + + + + + + + + C 2 + + + + + + + +C 3 + + + + + + + + C 4 + + + + + + + +C 5 + + + + + + + +C 6 + + + + + + + +C 7 + + + + + + + +C 8 + + + + + + + +C 9 + + + + + + + + C 10 + + + + + + + +C 11 + + + + + + + +C 12 + + + + + + + +

    C 13 + + + + + + + + C 14 + + + + + + + + C 15 + + + + + + + +

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    Classication of Projective Ring Lines up to Order 63

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    Line Cardinalities of Points RepresentativeType Rings

    Tot TpI 1N 2N 3N Jcb MD63/15 80 78 16 2 0 2 8 GF (7) GF (9)63/27 96 90 32 6 0 14 4 GF (7) [Z 9 or GF (3)[x]/ x 2 ]63/39 128 102 64 26 6 4 4 GF (7) GF (3) GF (3)62/32 96 94 33 2 0 29 3 GF (2) GF (31)61/1 62 62 0 0 0 0 62 GF (61)60/36 120 96 59 24 6 5 4 GF (3) GF (5) GF (4)60/44 144 104 83 40 12 15 3 GF (3) GF (5) [Z 4 or GF (2)[x]/ x 2 ]60/52 216 112 155 104 60 7 3 GF (3) GF (5) GF (2) GF (2)59/1 60 60 0 0 0 0 60 GF (59)58/30 90 88 31 2 0 27 3 GF (2) GF (29)57/21 80 78 22 2 0 16 4 GF (3) GF (19)56/14 72 70 15 2 0 1 8 GF (7) GF (8)56/32 96 88 39 8 0 23 3 GF (7) Z 8 , GF (7) GF (2)[x]/ x 3 , . . .56/38 120 94 63 26 6 17 3 GF (7) GF (2) GF (4)56/44 144 100 87 44 12 11 3 GF (7) GF (2) [Z 4 or GF (2)[x]/ x 2 ]56/50 216 106 159 110 66 5 3 GF (7) GF (2) GF (2) GF (2)55/15 72 70 16 2 0 6 6 GF (5) GF (11)54/28 84 82 29 2 0 25 3 GF (2) GF (27)54/36 108 90 53 18 0 17 3 GF (2) Z 27 , GF (2) GF (3)[x]/ x 3 , . . .54/38 120 92 65 28 6 15 3 GF (2) GF (3) GF (9)54/42 144 96 89 48 18 11 3 GF (2) GF (3) [Z 9 or GF (3)[x]/ x 2 ]54/46 192 100 137 92 54 7 3 GF (2) GF (3) GF (3) GF (3)53/1 54 54 0 0 0 0 54 GF (53)

    52/16 70 68 17 2 0 9 5 GF (13) GF (4)52/28 84 80 31 4 0 23 3 GF (13) [Z 4 or GF (2)[x]/ x 2 ]52/40 126 92 73 34 6 11 3 GF (13) GF (2) GF (2)51/19 72 70 20 2 0 14 4 GF (3) GF (17)50/26 78 76 27 2 0 23 3 GF (2) GF (25)50/30 90 80 39 10 0 19 3 GF (2) [Z 25 or GF (5)[x]/ x 2 ]50/34 108 84 57 24 6 15 3 GF (2) GF (5) GF (5)49/1 50 50 0 0 0 0 50 GF (49)49/7 56 56 6 0 0 6 8 Z 49 , GF (7)[x]/ x 2

    49/13 64 62 14 2 0 0 8 GF (7) GF (7)48/18 68 66 19 2 0 13 4 GF (3) GF (16)48/24 80 72 31 8 0 7 4 GF (3) [GF (4)[x]/ x 2 or Z 4 [x]/ x 2 + x + 1 ]48/30 100 78 51 22 6 3 4 GF (3) GF (4) GF (4)48/32 96 80 47 16 0 15 3 GF (3) Z 16 , GF (3) Z 4 [x]/ x 2 , . . .48/34 108 82 59 26 6 13 3 GF (3) GF (2) GF (8)48/36 120 84 71 36 12 11 3 GF (3) GF (4) [Z 4 or GF (2)[x]/ x 2 ]48/40 144 88 95 56 24 7 3 GF (3) Z 4 Z 4 , GF (3) GF (2) Z 8 , . . .48/42 180 90 131 90 54 5 3 GF (3) GF (2) GF (2) GF (4)48/44 216 92 167 124 84 3 3 GF (3) GF (2) GF (2) [Z 4 or GF (2)[x]/ x2 ]48/46 324 94 275 230 186 1 3 GF (3) GF (2) GF (2) GF (2) GF (2)47/1 48 48 0 0 0 0 48 GF (47)46/24 72 70 25 2 0 21 3 GF (2) GF (23)45/13 60 58 14 2 0 4 6 GF (5) GF (9)

    45/21 72 66 26 6 0 8 4 GF (5) [Z 9

    or GF (3)[x]/ x2

    ]45/29 96 74 50 22 6 2 4 GF (5) GF (3) GF (3)44/14 60 58 15 2 0 7 5 GF (11) GF (4)44/24 72 68 27 4 0 19 3 GF (11) [Z 4 or GF (2)[x]/ x 2 ]44/34 108 78 63 30 6 9 3 GF (11) GF (2) GF (2)

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    Line Cardinalities of Points RepresentativeType Rings

    Tot TpI 1N 2N 3N Jcb MD43/1 44 44 0 0 0 0 44 GF (43)42/30 96 72 57 24 6 11 3 GF (2) GF (3) GF (7)41/1 42 42 0 0 0 0 42 GF (41)40/12 54 52 13 2 0 3 6 GF (5) GF (8)40/24 72 64 31 8 0 15 3 GF (5) Z 8 , GF (5) GF (2)[x]/ x 3 , . . .40/28 90 68 49 22 6 11 3 GF (5) GF (2) GF (4)40/32 108 72 67 36 12 7 3 GF (5) GF (2) [Z 4 or GF (2)[x]/ x 2 ]40/36 162 76 121 86 54 3 3 GF (5) GF (2) GF (2) GF (2)39/15 56 54 16 2 0 10 4 GF (3) GF (13)38/20 60 58 21 2 0 17 3 GF (2) GF (19)37/1 38 38 0 0 0 0 38 GF (37)36/12 50 48 13 2 0 5 5 GF (4) GF (9)36/18 60 54 23 6 0 5 4 GF (4) [Z 9 or GF (3)[x]/ x 2 ]36/24a 80 60 43 20 6 1 4 GF (4) GF (3) GF (3)36/20 60 56 23 4 0 15 3 [Z 4 or GF (2)[x]/ x 2 ]GF (9)36/24b 72 60 35 12 0 11 3 [Z 4 or GF (2)[x]/ x 2 ] [Z 9 or GF (3)[x]/ x 2 ]36/28a 90 64 53 26 6 7 3 GF (2) GF (2) GF (9)36/28b 96 64 59 32 12 7 3 [Z 4 or GF (2)[x]/ x 2 ]GF (3) GF (3)36/30 108 66 71 42 18 5 3 GF (2) GF (2) [Z 9 or GF (3)[x]/ x 2 ]36/32 144 68 107 76 48 3 3 GF (2) GF (2) GF (3) GF (3)35/11 48 46 12 2 0 2 6 GF (5) GF (7)34/18 54 52 19 2 0 15 3 GF (2) GF (17)33/13 48 46 14 2 0 8 4 GF (3) GF (11)

    32/1 33 33 0 0 0 0 33 GF (32)32/11 45 43 12 2 0 0 5 GF (4) GF (8)32/16 48 48 15 0 0 15 3 Z 32 , GF (2)[x]/ x 5 , . . .32/17 51 49 18 2 0 14 3 GF (2) GF (16)32/18 54 50 21 4 0 13 3 GF (8) [Z 4 or GF (2)[x]/ x 2 ]32/20 60 52 27 8 0 11 3 GF (4) Z 8 , GF (2) GF (4)[x]/ x 2 , . . .32/23 75 55 42 20 6 8 3 GF (2) GF (4) GF (4)32/24 72 56 39 16 0 7 3 GF (2) Z 16 , Z 4 Z 8 , . . .32/25 81 57 48 24 6 6 3 GF (2) GF (2) GF (8)32/26 90 58 57 32 12 5 3 GF (2) GF (4) [Z 4 or GF (2)[x]/ x 2 ]32/28 108 60 75 48 24 3 3 GF (2) GF (2) Z 8 , GF (2) Z 4 Z 4 , . . .32/29 135 61 102 74 48 2 3 GF (2) GF (2) GF (2) GF (4)

    32/30 162 62 129 100 72 1 3 GF (2) GF (2) GF (2) [Z 4

    or GF (2)[x]/ x2

    ]32/31 243 63 210 180 150 0 3 GF (2) GF (2) GF (2) GF (2) GF (2)31/1 32 32 0 0 0 0 32 GF (31)30/22 72 52 41 20 6 7 3 GF (2) GF (3) GF (5)29/1 30 30 0 0 0 0 30 GF (29)28/10 40 38 11 2 0 3 5 GF (7) GF (4)28/16 48 44 19 4 0 11 3 GF (7) [Z 4 or GF (2)[x]/ x 2 ]28/22 72 50 43 22 6 5 3 GF (7) GF (2) GF (2)27/1 28 28 0 0 0 0 28 GF (27)27/9 36 36 8 0 0 8 4 Z 27 , GF (3)[x]/ x 3 , . . .27/11 40 38 12 2 0 6 4 GF (3) GF (9)27/15 48 42 20 6 0 2 4 GF (3) [Z 9 or GF (3)[x]/ x 2 ]27/19 64 46 36 18 6 0 4 GF (3) GF (3) GF (3)26/14 42 40 15 2 0 11 3 GF (2) GF (13)25/1 26 26 0 0 0 0 26 GF (25)25/5 30 30 4 0 0 4 6 Z 25 , GF (5)[x]/ x 2

    25/9 36 34 10 2 0 0 6 GF (5) GF (5)

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    Line Cardinalities of Points RepresentativeType Rings

    Tot TpI 1N 2N 3N Jcb MD24/10 36 34 11 2 0 5 4 GF (3) GF (8)24/16 48 40 23 8 0 7 3 GF (3) Z 8 , GF (3) GF (2)[x]/ x 3 , . . .24/18 60 42 35 18 6 5 3 GF (3) GF (2) GF (4)24/20 72 44 47 28 12 3 3 GF (3) GF (2) [Z 4 or GF (2)[x]/ x2 ]24/22 108 46 83 62 42 1 3 GF (3) GF (2) GF (2) GF (2)23/1 24 24 0 0 0 0 24 GF (23)22/12 36 34 13 2 0 9 3 GF (2) GF (11)21/9 32 30 10 2 0 4 4 GF (3) GF (7)20/8 30 28 9 2 0 1 5 GF (5) GF (4)20/12 36 32 15 4 0 7 3 GF (5) [Z 4 or GF (2)[x]/ x 2 ]20/16 54 36 33 18 6 3 3 GF (5) GF (2) GF (2)19/1 20 20 0 0 0 0 20 GF (19)18/10 30 28 11 2 0 7 3 GF (2) GF (9)18/12 36 30 17 6 0 5 3 GF (2) [Z 9 or GF (3)[x]/ x 2 ]18/14 48 32 29 16 6 3 3 GF (2) GF (3) GF (3)17/1 18 18 0 0 0 0 18 GF (17)16/1 17 17 0 0 0 0 17 GF (16)16/4 20 20 3 0 0 3 5 Z 4 [x]/ x2 + x + 1 , GF (4)[x]/ x2

    16/7 25 23 8 2 0 0 5 GF (4) GF (4)16/8 24 24 7 0 0 7 3 Z 16 , Z 4 [x]/ x2 , GF (2)[x]/ x 4 , . . .16/9 27 25 10 2 0 6 3 GF (2) GF (8)16/10 30 26 13 4 0 5 3 GF (4) [Z 4 or GF (2)[x]/ x 2 ]16/12 36 28 19 8 0 3 3 Z 4 Z 4 , GF (2) Z 8 , . . .16/13 45 29 28 16 6 2 3 GF (2) GF (2) GF (4)16/14 54 30 37 24 12 1 3 GF (2) GF (2) [Z 4 or GF (2)[x]/ x2 ]16/15 81 31 64 50 36 0 3 GF (2) GF (2) GF (2) GF (2)15/7 24 22 8 2 0 2 4 GF (3) GF (5)14/8 24 22 9 2 0 5 3 GF (2) GF (7)13/1 14 14 0 0 0 0 14 GF (13)12/6 20 18 7 2 0 1 4 GF (3) GF (4)12/8 24 20 11 4 0 3 3 GF (3) [Z 4 or GF (2)[x]/ x 2 ]12/10 36 22 23 14 6 1 3 GF (3) GF (2) GF (2)11/1 12 12 0 0 0 0 12 GF (11)10/6 18 16 7 2 0 3 3 GF (2) GF (5)

    9/1 10 10 0 0 0 0 10 GF (9)9/3 12 12 2 0 0 2 4 Z 9 , GF (3)[x]/ x2

    9/5 16 14 6 2 0 0 4 GF (3) GF (3)8/1 9 9 0 0 0 0 9 GF (8)8/4 12 12 3 0 0 3 3 Z 8 , GF (2)[x]/ x 3 , . . .8/5 15 13 6 2 0 2 3 GF (2) GF (4)8/6 18 14 9 4 0 1 3 GF (2) [Z 4 or GF (2)[x]/ x 2 ]8/7 27 15 18 12 6 0 3 GF (2) GF (2) GF (2)7/1 8 8 0 0 0 0 8 GF (7)6/4 12 10 5 2 0 1 3 GF (2) GF (3)5/1 6 6 0 0 0 0 6 GF (5)4/1 5 5 0 0 0 0 5 GF (4)

    4/2 6 6 1 0 0 1 3 Z 4 , GF (2)[x]/ x2

    4/3 9 7 4 2 0 0 3 GF (2) GF (2)3/1 4 4 0 0 0 0 4 GF (3)2/1 3 3 0 0 0 0 3 GF (2)

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    4. FINITE GENERALIZED QUADRANGLES

    Denition, Order, Grids and Dual Grids

    A nite generalized quadrangle of order (s, t ), usually denoted GQ(s, t ),is an incidence structureS = (P,B, I), where P and B are disjoint (non-empty) sets of objects, called respectively points and lines, and where I isa symmetric point-line incidence relation satisfying the following axioms:

    (i) each point is incident with 1 +t lines (t 1) and two distinct pointsare incident with at most one line;

    (ii) each line is incident with 1 +s points (s 1) and two distinct linesare incident with at most one point; and

    (iii) if x is a point and L is a line not incident withx, then there existsa unique pair (y, M ) P B for whichxIM IyIL; from these axioms itreadily follows that|P | = ( s + 1)(st + 1) and |B | = ( t + 1)(st + 1).

    It is obvious that there exists a point-line duality with respect to whicheach of the axioms is self-dual. Interchanging points and lines inS thusyields a generalized quadrangleS D of order (t, s ), called the dual of S .

    If s = t, S is said to have orders. A generalized quadrangle of order(s, 1) is called a grid and that of order (1, t ) a dual grid. A generalizedquadrangle with both s > 1 and t > 1 is called thick.

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    Some Distinguished Subsets: Perp-sets, Ovoids, Triads, etc.

    Given two points x and y of S one writes x y and says that x and

    y are collinear if there exists a lineL of S incident with both.

    For any x P denote x = {y P |y x} and note that x x;obviously, x = 1+ s + st . Given an arbitrary subsetA of P , the perp(-set)of A, A, is dened asA = {x|x A} and A := (A).

    A triple of pairwise non-collinear points of S is called atriad ; given anytriad T , a point of T is called its center and we say thatT is acentric,centric or unicentric according as|T | is, respectively, zero, non-zero orfeatures just a single point.

    An ovoid of a generalized quadrangleS is a set of points of S such thateach line of S is incident with exactly one point of the set; hence, eachovoid containsst + 1 points.

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    Geometric Hyperplanes and Associated Veldkamp Spaces

    The concept of crucial importance is ageometric hyperplane H of a point-

    line geometry (P, L ), which is a proper subset of P such that each line of meets H in one or all points.

    For = GQ(s, t ), one can check that H is one of the following threekinds:

    (i) the perp-set of a pointx, x;(ii) a (full) subquadrangle of order (s, t ), t < t ; and(iii) an ovoid.

    Finally, we introduce the notion of theVeldkamp space of a point-lineincidence geometry (P, L ), V ().V () is the space in which

    (i) a point is a geometric hyperplane of and(ii) a line is the collectionH 1H 2 of all geometric hyperplanesH of such

    that H 1 H 2 = H 1 H = H 2 H or H = H i (i = 1, 2), whereH 1 and H 2are distinct points of V ().If = S , from the preceding paragraph we learn that the points of V (S )are, in general, of three different types.

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    Smallest Generalized Quadrangle GQ(2,2) and Its Veldkamp Space

    The smallest thick GQ is obviously the one withs = t = 2, dubbed

    the doily. This quadrangle has a number of interesting representations of which we mention the most important two.

    One, frequently denoted asW 3(2) or simply W (2), is in terms of thepoints of P G (3, 2) (i. e., the three-dimensional projective space over theGalois eld with two elements) together with the totally isotropic lineswith respect to a symplectic polarity.

    The other, usually denoted asQ(4, 2), is in terms of points and lines of a parabolic quadric inP G (4, 2).

    From the preceding section we readily get thatW (2) is endowed with 15points/lines, each line contains three points and, dually, each point is onthree lines; moreover, it is a self-dual object, i. e., isomorphic to its dual.W (2) features all the three kinds of hyperplanes, of the following cardinal-ities:

    15 perp-sets,x, seven points each;10 grids (of order (2, 1)), nine points each; andsix ovoids, ve points each

    as depicted in the following gure.

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    Figure 2: The three kinds of geometric hyperplanes of W (2). The points of the quadrangle are represented by smallcircles and its lines are illustrated by the straight segments as well as by the segments of circles; note that not every

    intersection of two segments counts for a point of the quadrangle. The upper panel shows the points perp-sets (yellowbullets), the middle panel grids (red bullets) and the bottom panel ovoids (blue bullets); the use of different colouringwill become clear later. Each picture except that in the bottom right-hand corner stands for ve differenthyperplanes, the four other being obtained from it by its successive rotations through 72 degrees around the center of the pentagon.

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    The quadrangle exhibits two distinct kinds of triads, viz. unicentric andtricentric.

    A point of W (2) is the center of four distinct unicentric triads (Figure,left); hence, the number of such triads is 4 15 = 60.

    Tricentric triads always come in complementary pairs, one represent-ing the centers of the other, and each such pair is the complement of a gridof W (2) (Figure, right); hence, the number of such triads is 2 10 = 20.

    A unicentric triad is always a subset of an ovoid, which is never the casefor a tricentric triad; the latter, in graph-combinatorial terms, representinga complete bipartite graph on six vertices.

    Figure 3: Left : The four distinct unicentric triads (grey bullets) and their common center (black bullet); note thatthe triads intersect pairwise in a single point and their union covers fully the centers perp-set. Right : A grid (redbullets) and its complement as a disjoint union of two complementary tricentric triads (black and grey bullets); thetwo triads are also seen to comprise a dual grid (of order (1 , 2)).

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    Now, we have enough background information at hand to reveal the struc-ture of the Veldkamp space of our doily,V (W (2)).

    From the denition, we easily see thatV (W (2)) consists of 31 points of which 15 are represented/generated by single-point perp-sets, 10 by gridsand six by ovoids.

    The lines of V (W (2)) feature three points each and are of ve distincttypes, as illustrated in Figure.

    From the properties of W (2) and its triads as discussed above it read-ily follows that the number of the lines of type one to ve is 60, 20, 15, 45and 15, respectively, totalling 155.

    Hence, V (W (2)) has the same number of points (31) and lines (155) asP G (4, 2), the four-dimensional projective space over the Galois eld of twoelements; this is not a coincidence, as the two spaces are, in fact, isomorphicto each other.

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    Figure 4: The ve different kinds of the lines of V (W (2)), each being uniquely determined by the properties of itscore-set (black bullets). Note that the yellow hyperplanes (i. e., perp-sets) occur in each type, and yellow is also thecolour of two homogeneous (i. e., endowed with only one kind of a hyperplane) types (2nd and 3rd row). It is also

    worth mentioning that the cardinality of core-sets is an odd number not exceeding ve. The three hyperplanes of anyline are always in such relation to each other that their union comprises all the points of W (2).

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    GQ(2,2) and the Projective Line over the Full Two-by-Two Matrix Ring with Entries in GF(2)

    Already shown.There exists a perfect parity between the three kinds of the geometric hy-perplanes of the generalized quadrangle of order two and the three kinds of the projective lines over the rings of four elements and characteristic twoembedded in our sub-conguration of P 1(M 2(GF (2))), as summarized inTable.

    GQ Ovoid Perp-Set\{ X } GridPL P 1(GF (4)) P 1(GF (2)[x]/ x2 ) P 1(GF (2) GF (2))

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    GQ(2,2) and Icosahedron

    Sylwesters Syntheme-Duad Geometry on a Six-Letter Set

    A duad is an unordered pairij , i = j with i, j {1, 2, 3, 4, 5, 6}

    A syntheme is a set of three duads{ij,kl,mn } partitioning the six-letterset, i.e. with i,j ,k,l ,m,n being distinct.

    Sylwesters syntheme-duad geometry with duads as points and synthemesas lines is ismorphic toGQ(2, 2).

    Icosahedron a Platonic solid with 30 edges, 20 faces and 12 verticesA representation of GQ(2, 2) on the icosahedron:

    Six-letter set Set of six axes connecting opposite vertices (gure left)

    Points of GQ(2, 2) as pairs of opposite edges of the icosahedron (gure right)

    Figure 5:

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    One can identify of the three kinds of a hyperplane of GQ(2, 2) with thethree kinds of a symmetry axis of the icosahedron, namely

    per-sets with two-fold (edges),

    grids with three-fold (faces), andovoids with ve-fold (vertices) symmetry axes (gure).

    Figure 6:

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    5. References

    Blunck A, Havlicek H. Projective representations I: Projective lines overa ring. Abh Math Sem Univ Hamburg 2000;70:28799.

    Havlicek H. Divisible designs, Laguerre geometry, and beyond. Quadernidel Seminario Matematico di Brescia 2006;11:163. A preprint availablefrom: http://www.geometrie.tuwien.ac.at/havlicek/dd-laguerre.pdf .

    Veldkamp FD. Geometry over rings. In: Buekenhout F, editor. Hand-book of incidence geometry. Amsterdam: Elsevier; 1995. p. 103384.

    Herzer A. Chain geometries. In: Buekenhout F, editor. Handbook of incidence geometry. Amsterdam: Elsevier; 1995. p. 781842.

    Saniga M, Planat M, Kibler MR, Pracna P. A classication of the pro- jective lines over small rings. Chaos, Solitons & Fractals 2007;33:1095-1102.

    Payne SE, and Thas JA. Finite Generalized Quadrangles. Pitman:

    BostonLondonMelbourne; 1984. Buekenhout F, and Cohen AM. Diagram Geometry (Chapter 10.2).

    Springer: New York; 2007 (to appear). Preprints of separate chapterscan be found at http://www.win.tue.nl/ amc/buek/

    Ronan MA. Embeddings and hyperplanes of discrete geometries. Eu-ropean J Combin 1987;8:179185.