A METRIZATION FOR POWER-SETS AND CARTESIAN PRODUCTS TOTH
APPLICATIONS TO COMBINATORIAL ANALYSIS
DISSERTATION
Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University
ROBERT SILVERMAN, B. Sc., M. A.
******
The Ohio .State University 1958
Approved by
Adviser Department of Mathematics Acknowledgements
®he author wishes to express his sincere appreciation to
Professors Marshall Hall, Jr. and Henry 5. Mann for their reading of the dissertation and their suggestions, and especially to his adviser, Professor D. Hansom Whitney, to Professor Herbert J. Terser, and to his wife, Donna, for their continued encouragement and assistance.
ii Contents
Section Page
1. Introduction ...... 1
2. Abstract metric spaces ...... 1
3. A metrization for power-sets and Cartesian products .• U
U> The metric space 5^(n) ...... 5
5. Metric characterizations of some classical
configurations and their generalizations ...... 6
6. Some theorems for the metric space S^(n) ...... 18
7. Future research ...... 31
Bibliography...... 33
Autobiography...... 35
iii 1. Introduction. Virtually every combinatorial configuration may be phrased in terns of certain conditions on families of subclasses of the power-set Ug (the class of all subsets of S) of some abstract set S. In view of this, it would seem worthwhile to investigate systematically some aspects of the behavior of
Ug. In this paper a method of structuring the power-set is proposed /by defining a distance function over Ug. Since
Cartesian products may be viewed as subsets of power-sets, this definition also leads to a metrization for such sets. The present paper is concerned primarily with an examination of some
of the metric properties of Cartesian products and their relation to some of the classical configurations and their generalizations.
2. Abstract metric spaces. In order to reduce to a minimum the introduction of new terminology, wherever feasible the author has adopted that used by ELumenthal [Ul [f>l, whose excellent books
also nourished several interesting trains of thought.
DEFINITION 1. Given a set M, an r-dimensional metric over M, r > 2, is a mapping which associates with every ordered r-tuple
a]_,a2, ...,ar of elements of M, a non-negative real number d(a^,...,ar), termed the distance of the points a^, ...,ar,
satisfying:
M l . d( a^,..., &p) > 0.*
M 2. d(a-j_, ...,ar) = 0 if and only if a-^ = a2 s ... = ar.
1 2
M 3 . (Symmetry) If • • .>ir is any permutation of the integers 1,2,..., r, then d( ap ..., ar) = d(a^, . . a ^ ) .
M U« (Triangle inequality) For every set of r + 1 points
) «• •» aj. + ]_ of M,
d(a^ *«• >ar) < ? d(a^, ...,a^ _ + ^>..»,ar + ^)»
The postulate M 2 differs somewhat from that in ELumenthal
[U, P* 126], but is more suitable to our purposes. Also, in keeping with the progressive spirit of the times, our "dimension” is one higher than the comparable "dimension” in ELumenthal.
In the applications there frequently will be metrics of various dimensions defined over M with connecting conditions. Generally we will use the symbol M to denote either the set M, or the metric space consisting of M and the metric d over M. Wien it is desirable to differentiate explicitly the two concepts, we will symbolize the latter by (M,d).
We now define some concepts for conventional (two-dimensional) metric spaces. In most instances the extension to r-dimensional spaces is obvious. In the following, M denotes a metric space, and E a subspace of M.
DEFINITION 2. If M ■ £a-^,a2, ...j is countable, it may be completely specified by the symmetric distance matrix
A - (ai;j)> ai;j ■ d(a±,a3). 3
DEFINITION 3. For a in M, r > 0, the open sphere aid closed sphere with center a, radius r are defined, respectively, by
Note that in general a sphere need not have a unique center or radius.
DEFINITION li. The major diameter A(E), of E, and the minor diameter /( E), of E are defined by
A(E) ■ l.u.b. d(x,y) for x,y in E,
/(E) = g.l.b. d(x,y) for x,y in E, x y.
If E contains fewer than two points, define /(E) = 0.
DEFINITION E>. The t-extent of E, e(E,t), is the greatest integer m such that E contains m distinct points with minor diameter greater than t. If no two points of E have distance greater than t, set e(E,t) * 1, while if for n arbitrarily large there are n points of E with minor diameter exceeding t, define e(E,t) = + °o.
DEFINITION 6. Two metric spaces M and M' are isometric provided there exists a mapping a from M onto M' such that d(x,y) » d(a(x),a(y)) for all x,y in M. We write M * M'.
Note that a is biunique since a(x) ■ a(y) implies d(a(x),a(y)) * d(x,y) “ 0. If M = M', the isometry is termed a motion. Two subsets of M are superimposable provided a motion exists that maps one onto the other. u
DEFINITION 7. E is a metric basis of M provided each point of M is uniquely deteimined by its distances from the points of E. The basis is irreducible provided no proper subset of E is a metric basis, and is minimal provided M has no metiic basis of smaller cardinality. Finally, the basis is complete provided for each subset E' of M with E “ E', and each subset T' of M containing E", a subset T of M exists such that the isametxy of E with E' can be extended to T S T'. The extension is obviously unique. Also, note that for M finite, for completeness it is sufficient to require simply that E and E' be superimposable for each subset E' of M with E ~ E'.
DEFINITION 8. For x in M, the distance of x from E is given ty d(x,E) » g.l.b. d(x,y) for y in E.
3. A metrization for power-sets and Cartesian products.
Given a set S, let Ug denote its power set. For A in Ug, let
N(A) be the number of elements in A, where + ®o is an admissible value. For A^, •••,Ar in Ug define:
DEFINITION 9. d(A!,...,Ar) - N(\JA± - A A i ) / r .
THEOREM 1. Under the above definition, d is an r-dimensional metric over Ug.
Proof. M 1, M 2, M 3 are immediate. Let A^,...,Aj, + be in Ug, Let a L ° d(A]_, ...,Aj.), R ** 2 d(A]_, .• Aj_ «• i>Aj_ + x* • ••jAp + x^* <■'=/ 5
If R = + «o , then certainly L < R, so we may assume R < + oo .
Let x be arbitrary in S. Then x can contribute 0 to R if and
only if x is either in all of the r + 1sets or innone. In
either case it also contributes 0 to L. In all other cases
x contributes at least l/r to R and at most l/r to L. Hence
L < R. Thus M U is satisfied.
In particular, definition 9 defines an r-dimensional metric
for Cartesian products as follows: Given the sets S , ...,,S , X Jx denote by n Si ° Si x S 2 x ... x Sjj ■ ^(s^, ...,sk)» s^ in
their Cartesian product of ordered k-tuples. Then it may be
regarded as a subclass of Ug(k ) x where 2, ...,kj,
by regarding the element (a^,. .^a^) of n as the subset
|(l,a1),...,(k,ak )j of S(k) x
k* ^*1° metric space S^(n). Let S(n) denote a set with
n elements, n > 1. Unless otherwise indicated, S(n) will be
taken as the first n positive integers. We denote by S^-(n), k > 1,
the k-fold Cartesian product of S(n),
S ^ n ) - {(x l , ...,xk); X£ in S(n)j, and assume
that S^(n) has been metrized as in the preceding section.
Although it appears that higher dimensional metrics will be
necessary to characterize such items as the theorem of Desargues,
in the present paper we confine our attention to the two-
dimensional metric, which for this special case we may restate: For x,y in S^n), x - ( x ^ . . . ^ ) , y - (y-p ...,yk ), d(x,y) is the number of subscripts i for which x^ / y^, i = 1, ...,k.
For 0 < r < k, every set of nr + 1 elements of Sk (n) has minor diameter at most k-r. Hence the k-r extent of every subspace E of S^(n) satisfies e(E, k - r) < nr . We next define terms to describe subspaces which attain this maximum extent.
DEFINITION 10. A subspace E of S^(n) is r-orthogonal,
0 < r < k, provided e(E, k -r) = nr. If in addition E contains precisely nr elements, E is termed an L(n,k,r) space. (Thus
E is r-orthogonal if and only if E contains an L(n,k,r) space.)
For a given subspace E, r-orthogonality does not imply
(r-1) -orthogonality. S^(n) always has orthogonality 0,1 and k.
Indeed, any point constitutes an L(n,k,0) space/ the points
(i,i,...,i), i » 1,...,n comprise an L(n,k,l) space, and
S^(n) is itself an L(n,k,k) space. For values between 1 and k the property becomes non-trivial, and, as we shall see in the following section, is related to some of the classical unsolved problems in combinatorial analysis.
5. Metric characterizations of some classical configurations
and their generalizations.
a) LATIN SQUARES AND CUBES. A Latin square of order n,
A = [ay], is an n x n matrix whose entries are from a set of n distinct symbols and such that each symbol appears exactly 7 once in each row and column. Thus a Latin square of order n is essentially the multiplication table of a loop of order n.
Two Latin squares A " (aij)j B = (^ij) of order n are Greco-Latin provided the n^ ordered pairs (a^j, b ^ ) are all distinct.
A set of Latin squares of order n, A^,A2, . ..jA^ are orthogonal provided Ai and Aj are Greco-Latin for all i / j. In this event,
one readily shows that m < n - 1 . An orthogonal set is complete provided m ■ n - 1 .
A Latin cube of order n, A * ta^^], is a cubical array
of n^ cells (in n row-planes, n column-planes, and n layers) whose entries are from a set of n distinct symbols and such
that whenever a y ^ • auvw and at least two of the equalities
r = u, s = v, t = w hold, then the third also holds. Note that
this condition holds if and only if each row-plane, column-plane,
and layer is a Latin square of order n.
Two Latin cubes of order n are Greco-Latin provided every
pair of corresponding row-planes, column-planes, and layers is
a Greco-Latin square. Three Latin cubes, A * B “ ^ijk-k
C = [cijk], of order n are strongly Eulerian provided each pair
is Greco-Latin and the n^ ordered triples (a , b , c ,,) ijxc l j K 3.j k are all distinct. (These conditions are stronger than those
of Ball [2]. ) A set of Latin cubes of order n, A p ...,Am ,
are orthogonal provided A^, A^ A ^ are strongly Eulerian for all
i, pairwise distinct. Again one readily shows that m < n - 1, 8 and an orthogonal set is termed complete provided m * n - 1 .
The literature on Latin squares appears to be almost as extensive as that on magic squares. (In this connection, see the excellent historical review in Norton's paper [20] ).
Dudeney [12] poses a card puzzle which he believes to have been
first published in the 162U edition of the work of Bachet de Meziriac, and which asks in how many ways the sixteen court
cards (including the aces) can be arranged in a square array
"so that in no row of four cards, horizontal, vertical or
diagonal shall be found two cards of the same suit or the same
value." It is evident that this problem is equivalent to
enumerating the number of Greco-Latin squares of the fourth
order having the additional condition on the diagonals. Euler,
however, is generally credited with first focusing the serious
attention of the mathematical world on the Latin square in his
"Recherches sur un nouvelle espece de quarres magiques." In
1799 he proposed the Problem of the Thirty-Six Officers:
"A meeting of 36 officers of 6 different ranks and from 6 different
regiments must be arranged in a square in such a manner that
each orthogonal contains 6 officers from different regiments and
of different ranks." The problem is equivalent to constructing
a Greco-Latin square of order 6, and Tarry [26] showed this to
be impossible by a clever enumeration method.
Euler conjectured that for n * [4k + 2, Greco-Latin squares
of order n do not exist. Aside from n ■ 6 (n = 2 is, in a sense, 9 vacuous since a complete set consists of a single square), the question of the validity of the conjecture has resisted all determined onslaught (although Mann [18] has ruled out certain candidates, among these being the group multiplication tables).
The case n = 10 remains the first undecided instance. MacNeish
[17] seems to have been the first to establish the existence of complete sets of orthogonal Latin squares of prime power order. The interest in orthogonal Latin squares and finite projective planes was mutually enhanced when Bose [6] and
Levi [16] independently showed the equivalence of complete sets of such squares to the planes.
Given a set of orthogonal Latin squares A^, ...jA^ of 2 order n, construct the associated n k-tuples (k « m + 2) in the usual manner. (Here the k-tuple (i^,...,i, ) is admitted if and only if i. is in row i , column i, of Aj, j = 1, ...,k-2.) v k-1 J o One readily verifies that these n elements actually comprise an L(n,k,2) space, since any pair of the elements having at least two corresponding components equal would violate either the Latin condition on rows or columns, or the orthogonality 2 condition. Thus the minor diameter of the n elements exceeds k - 2. Conversely, given an L(n,k,2) space, we reverse the process and obtain k - 2 orthogonal Latin squares of side n. The same procedure may be employed to show the equivalence of L(n,k,3) spaces and sets of k - 3 orthogonal Latin cubes. 10
b) FINITE NETS. For k,n positive integers with k > 3,
Brock [9 ] defines a (finite) net N of degree k, order n, as
a system of undefined objects called ''points” and "lines" together with an incidence relationship "point is on line" or "line passes through point") such that: (i) N contains k (non-empty) classes of lines. (ii) Two lines n,b of N belonging to distinct classes, have a unique common point P. (iii) Each point P of N is on exactly one line of each class. (iv) Some line of N has exactly n distinct points. ...A finite affine (or Euclidean) plane with n points on each line, n > 2, is simply a net of degree n + 1, order n [lq], A loop of order n is essentially a net of degree 3, order n [l] [3 ]. More generally, for 3 < k < n + 1, a set of k - 2 mutually orthogonal n x n Latin squares may be used to define a net of degree k, order n (and conversely) by paralleling Bose's correspondence [6] between affine planes and complete sets of orthogonal Latin squares.
For k > r > 0 we may generalize Gruck’s configuration to a finite net of degree k, order n, and dimension r by replacing
(ii) and (iv) with
(ii") Every r lines 1^, ...,lr of N belonging to pairwise distinct classes have a unique common point P.
* Y — X (iv') Some line of N has exactly n points.
As immediate consequences of the axioms we have
1) Every class contains n lines.
2) Every line has exactly nr “ ^ points.
3) N contains nr points.
For let Ap...,Ar,Ar + p ...,Ajj. be the k classes of lines of N, and suppose contains m^ lines, and N contains m points. 11
Then from (ii') and (iii) we obtain the system of r + 1 equations n lttj ■ m (j * 1, . ..r + 1, j/*i»i®l, ...,r + l) which have the unique solution m^ = m1/1*, i “1,.. .r + 1, and since we may replace A^. + -j_ by any other Aj, we obtain m^ « mr^r, i ** 1, ...,k.
If we next consider any fixed line of Ai together with the classes A2>«».*Ar, then (ii') and (iii) imply that the line passes through m^r ~ ^ / T points, and this together with (iv') implies m = nr .
Now let us coordinatize N by assigning to the point P the coordinates (i^,...ji^) provided P is on the i^-th line of the j-th class. Then (ii') and (iii) imply that there is a
1-1 correspondence between points and coordinates, and that as elements of S^(n) any two of these ordered k-tuples have distance exceeding k-r. Since there are nr distinct such k-tuples, and each component of a k-tuple can assume n values, the nr k-tuples comprise an L(n,k,r) space. Conversely, one may reverse the above process, and we thus have a correspondence between L(n,k,r) spaces with 0 < r < k and finite nets of degree k, order n, dimension r. In particular then, an L(n,3,2) space is essentially a loop of order n, an L(n,n + 1,2) space defines a finite affine plane with n points on each line (n > 2), and from an L(n,k, 2) space we may construct a system of k - 2 orthogonal Latin squares of side n. 12
c) HYPERCUBES AND ORTHOGONAL ARRAYS. Rao [22] defines a hypercube of strength d as follows.
Let there be m factors A-oA^ each of which can assume s different Values. We define an ordered set (ip,i2* . ..,im ) as a combination of m factors obtained ty the selection of i^-th, i2_th... values of the first, second, ..., factors respectively. There are s such combinations of which a subset of st combinations may be called a (m,s,t) array. An (m, s,t) array is said to be of strength d if all combinations of any d of the m factors occur an equal number (s^ " <*) of times. An array of strength d represented by (m,s,t,d) is, alternatively, called a hyper cube of strength d#
For t = d, these bypercubes correspond to L(n,k, r) spaces with n = s, r = d, and conversely.
Bose and Bush [71 [8] weakened the condition of s^ combinations to H = As** combinations, to obtain an orthogonal array of strength d,
size N, index X, k constraints and s levels which they define as
na k x N matrix A, with entries from a set 2 of s > 2 elements...
[such that] each d x N submatrix of A contains all possible
d x 1 column vectors with the same frequency X." For X = 1
it is d e a r that the column vectors of A comprise an L(n,k,r)
space, and conversely.
d) A CONFIGURATION. Let v elements be arranged into
v + 1 sets T, T]_, ...,TV such that the number of elements which
are in T and T^ but not in both is k, i * 1, ...,v, and such
that for i f tj, the number of elements which are in either T^
or Tj but not in both is q. We may coordinatize the sets of the 13
configuration by assigning to a set the coordinates (i^,...,1^), where i^ = 1 if the j-th element is in the set, and i^ * 0
otherwise. If x,Xp ...xv are the coordinates of T,T^,. • T^, then x,x^, ...,xv comprise a subspace of Sv (2), S(2) = {o,lj,
satisfying
(i) d(x,x^) » k, i ** 1, ...,v
(ii) d C ^ X j ) - q, i £ j.
We will discuss this configuration further in the next example*
As an illustration, for k = 3, q = U, v = 7, consider
T-{l,2j T3-{l,U,6{ T6 = { i ,2,3,5,6
% " { 3 } -{ 2,6,7 } T? -{l,2,3,U,7
T2 = {2 > U , T 5 ={1,5,7}
e) THE v,k, A CONFIGURATION. Consider next the now classic
v,k,A configuration defined in Chowla and Hyser [ll] as an
arrangement of v elements into v sets such that every set
contains exactly k distinct elements and such that every pair
of sets has exactly A elements in common, 0< A < k < v. In
statistics these configurations are termed symmetrical balanced
incomplete block designs. For A * 1 and k ■ n + 1, n > 2, the
configuration reduces to a projective plane with n + 1 points
per line, and for v = Ijm - 1, k ■ 2m - 1, A = m - 1, it is
equivalent to a Hadamard matrix of order N = Ijm [21 ] (these are m the + 1 matrices H satisfying HH = NT, where H is of order
N and I is the identity matrix). For a comprehensive summary lit of results see Ityser [23]. With the v,k,X configuration we may associate its characterizing v x v incidence matrix A =(a ^ ), where a.^ ■ 1 if the j-th element is in the i-th set, and
0 otherwise. Actually, constructing the incidence matrix is equivalent to coordinatizing the sets of the configuration, the i-th row representing the coordinates of the i-th set.
It is apparent that the v sets of coordinates so obtained comprise v elements of Sv(2), S(2) = £o,l^, satisfying:
(i) If s * (0,0,...,0), and the v elements are x^,...jX^, then d(xps) = k for i = 1, ...,v*
(ii) d(xjL,x;j) = 2(k - X), i ^ j.
That the metric characterization of v,k,X tends toward the heart of the matter is suggested in (ii) by the fact that X is essentially ignored, whereas the value k - X which appears in both v,k,X and its complementaiy design (the design obtained by replacing each set by its complement) and plays such a critical role in the non-existence theorems, occurs explicitly.
Also of some interest is the fact that the ’’strong'1 converse of the above holds. That is, given v + 1 elements sA,x',...,x^ of Sv (2) satisfying
(iA) d(x£,sA) = k for i “ 1,...,v
(ii) d(x£,xp » 2(k - X), i / j, 0 < X < k < v, we may construct v elements X p ...jXy satisfying (i) and (ii) and hence constituting a v,k,X configuration. This may be seen 15 as follows. If in the j-th components of x^,. ..^x^s' we replace
0's by l ’s and l's by 0 ’s, we clearly obtain an isometric space.
Now perform this replacement in the j-th components if and only if the j-th component of s' is 1. Then we obtain an isometric space with s = (0,0,...,0) as the image of s'.
Consider again the admittedly contrived configuration in d).
Though, at least on the surfacd, the relation of this configuration to the v,k,X configuration is somewhat obscure, by relating both to their metric characterizations, we now see immediately that for k < v, 0 < q < 2k, they are essentially equivalent.
f) BALANCED INCOMPLETE BLOCK DESIGNS. Let T = ..., syj, and consider the configuration C *•£ Tp . where the Tj_ are subsets of T. Then the dual configuration consists of the subsets A p •.., Ay. of the set A = £tj..., t^J, where t^ is in A-j if and only if a^ is in T^» and the complementary configuration consists of the sets T p , ,Tp where T^ denotes the complement of T± .
Given the set S(b) of b elements, let Er denote the class of all subsets of S(b) containing r elements. Then Ep is a subspace of the two-dimensional metric space We will show that the existence of a balanced incomplete block design
(BIBD) [29] with parameters b,v,k,r,A (0 < A < r < b) is equivalent to having e(Er, ^t - lj) > v, where denotes the least integer > m and t ® rv(b - r)/b(v - l) (0 < r < b, rv > b). Note that if in the 16 latter equality, we define k * vr/b, \ » r(k - l)/(v - l)
(relations which always hold for the BIBD), then we have simply t = r - X.
THEORM 2. If eCEp, Jt - lj) ■ v' > v > 1, where t = t » rv(b-r)/b(v - l) and 0 < r < b, then v' = v and there exist v elements x^, ...,xv in Er such that dfe^x^) * t for all i f j.
Proof, Ey hypothesis there exist v' elements x^, .. in Ep, v' > v, such that d(xi,x;j) > t for all i / j. Let denote the number of sets containing the i-th element of
JS(b), i = 1, ,..,b. (kmputing total occurrences we obtain
(1) 2 k, = rv'. ‘SI . . Comparing contributions to all J set intersections, we obtain
(2) 2 = 2 NCxj^TNXj) » 2 tr-d(xi,x;j) 3.
But then d(x^x^) > t implies
O ) % (1?j < (r - t) ( l j .
2 2 2 Now by lemma 1 (section 6), (l) implies that r v' < b2 k^ , and so from (l) and (3) we obtain
(U) L - (rv'Krv' - b)/2b < M = 2 (g1] < R => (r - t) (lj.
Using 0 < r < b, v > 1, and the appropriate expression for t, we find from elementary calculations that L < R is equivalent 17 to v' < v, and L = R if and only if v' = v. Hence v' = v, and L = M = R. Finally, using (2), % [r - d(x^,x^) ] « (r - t) ^ 2 ^ * and thus d(x^,Xj) > t implies r - d(x^,Xj) = r - t. Hence d(x^,x^) = t, completing the proof of the theorem.
COROLLAS? 1. If rv > b, then the configuration x^, ...,x^_ is the dual of a BIHD with parameters b,v,k,r,X (0 < X < r < b), where X ■ r - t * r(vr - b)/b(v - 1) = r(k - l)/(v - 1) and k » rv/b. Conversely, given the BIHD, one can construct the dual configuration, and so obtain eCEp, £ t - ij) =» v.
Proof. From the condition rv > b we obtain the restriction
0 < X < r, thus eliminating trivial configurations. Next observe that in the above proof, M = R together with lemma 1
(section 6) and (1), implies k^ = rv/b * k. Thus every element of S(b) is replicated k times. Finally, note that d(x^,x^) = t for all i / j implies that every pair of distinct sets intersect in exactly X = r - t elements.
Specializing to v,k,X configurations, one obtains the interesting result:
COROLLARY 2. If v elements are arranged in v' > v sets such that every set contains exactly k distinct elements,
1 < k < v, and such that every pair of distinct sets has at most
X elements in common, where X = k(k - l)/(v - l) (X not necessarily integral) , then v' = v, every pair of distinct sets has exactly 18
X elements in common, and thus the arrangement constitutes a v,k,X configuration.
6. Some theorems for the metric space S^(n). Let denote a subspace of the two-dimensional metric space S^(n^), i = 1,...,t.
For » (a^, ...,aijc) in E^, i * 1, ...,t, let x ^ X g . . ^ = u Xj, ** (b^, •».,bj^), where bj - • 3 = 1,«»«,k» If n 3j_ ° (b^, ,..,1^), n » (c^,..., Cjj), x^ and y^ in E^, define d(n x^, n y^) in the usual manner as the number of subscripts j for which b^ / c^, j = 1, ...,k.
DEFINITION 11. The metric space
Ej_ 3: E2 x ... x Ej. “ n x^« x^, in E^J is termed the direct product of the E^. Note the distinction between the direct product n and the Cartesian product n
From the above definition, it is clear that any biunique mapping from n S(n^) onto S(n n^) induces an isometiy between n S^(n^) and S^(n n^). For convenience we simply write
= S^(n r_j_). With this understanding, n E^ is a subspace of ,S^(n ni). Note that the direct product is independent of the order of the factors. That is, if ip...,!^ is a permutation of 1,2, ...,t, then n E^ is isometric to rt Ej_ under the mapping n x . — *» Ji x. . Next note that ift,
Finally, note that under the correspondence
x± -*• x' - e1e2...ei _ ^ where e^ is a fixed element of E^, E^ isometric to the subspace of n Ej defined by E[ = ^x£/x^ in E^j .
THEOREM 3. For Xj_ and y^ in Ej,
max [d (xi,y±) ] < d (n x±, n y±) < % d(xjL,yi).
Proof. From the definition it is clear that if x and y rr r have distinct j-th components, then so do it x^ and it y^. Hence d(xr,yr) < d(n x^, n y^) and the first inequality follows.
Next suppose it x^ and n y^ have distinct j-th components. Then so do at least one pair and from this it is clear that the second inequality must hold.
COROLLARY 1. The major and minor diameters satisfy
max ( Proof. From the definitions and the theorem we have COROLLARY 2. If S^n-^) and S^ng) are r-prthogonal, then so is S^n^ng). Proof. Let L^ be an L(n^,k,r) space of S^n^), i =1,2. # 2c Then L = L^ x Lg is an L(n-jng,k,r) space of S (n^ng), since (n^ng)r elements. LEMMA 1* For the real numbers a^, •••an> (2 ai)^ < nZ a^, and equality holds if and only if naj s Z aj for all j. LEMMA 2. Let i^, ...,i^., t < r, be any t distinct integers from among 1, ...,k, and let a-? be in S(n). Then in the 3 L(n,k,r) space, L, there are precisely n elements with ij-th component equal to a^^, j = 1, ...,t. Proof. The proof is evident from the fact that L contains nr elements, any two distinct elements agree in at most r - 1 corresponding components, and over S(n) every t-tuple can be completed to an r-tuple in exactly nr “ ^ ways. LEMMA 3. If S^-(n) is r-orthogonal, then ^(n) is (r - l)-orthogonal* Proof. Let L be an L(n,k,r) space of S^(n). Py lemma 2 there are n elements in L having the same k-th component* If their k-th components are dropped, it is easy to see that the nr “ elements so obtained constitute an L(n,k - 1, r - l) space. ce. 21 LEMMA it. If S^ n ) is r-orthogonal, then so is S^(n), t < k. Proof. This is clear from the fact that if the last k - t components of the elements in an L(n,k,r) space are dropped, the resulting elements comprise an L(n,t,r) space. LEMMA 5. If S^(n) is r-orthogonal, r > 2, then k < n + r - l. Proof. Let L be an L(n,k,r) space in ,S^(n). Let x^,...,2^ denote the n elements of L with first r - 1 components equal to a, and let ^ + ^ be an element of L distinct from these and having its first r - 2 components equal to a, a in ,S(n) (lemma 2). Then it follows that d(xj,Xj) > k - r for all i / j implies that n > k - r + 1. LEMMA 6. For x and y in the L(n,n + r - 1, r) space, L, d(x,y) < n + 1 implies d(x,y) = n. Proof. Let x = (ai, ...ja^), y - (t^, ...jb^), k => n + r - 1. Suppose d(x,y) < n + 1. Then a^ = b^ for at least r - 2 subscripts i. With no loss of generality, suppose a^ = hj_, i ■ 1, ...,r - 2. Now by lemma 2 there are exactly n - 1 elements in L(n,k,r) which are different frc»a x and have i-th component equal to a^, i = 1, ...r -2. Let Aj denote the subset of these n^ - 1 elements having J-th component equal to a^, j « r - 1, ...,k. Then again by lemma 2, Aj contains precisely n - 1 elements. Also, A^^> = $ § o for i f J, and so U A j contains precisely (n - l) (n + 1) “ n - 1 22 elements. Thus every element in L having i-th component equal to a^, i = 1, ...,r - 2, has its j-th component equal to aj for precisely one value of j, r - 1 < j < k, and so has distance n frcra x. In particular, d(x,y) *= n. LEMMA 7. Given a k x r matrix, A, over a field F, having all its r-rowed minors nonsingular, we can construct a k x (k - r) matrix with all (k - r)-rowed minors nonsingular. Proof. Let A^ denote the r x r matrix consisting of the first r rows of A, and let denote the (k - r) x r matrix consisting of the remaining k - r rows, so that A * I ^ I . W By hypothesis Ar, is nonsingular, so by elementary operations on the columns where I is the r x r identity matrix. Since the elementary operations have been performed only on the columns of A, A' also has all r-rowed minors nonsingular. Now using the Laplace expansion, one sees that every minor determinant of A^ occurs as a factor of some r-rowed minor determinant of A', and hence is not zero. Again applying the Laplace expansion, one verifies that every (k - r)-rowed minor determinant of the (k - r) x k matrix [A', 11 either equals unity or has the same absolute value as some minor determinant of A' (here I is the (k - r)-rowed identity matrix). Hence [Agji] has all (k-r)-rowed minors nonsingular. Taking the transpose, we have the required result. 23 THEOREM k- Given a k x r matrix A = [a^ 1 over GFCp3®) having all r-rowed minors nonsingular, we can construct an L(n,k,r) and an L(n,k,k - r) space, n * pm . Proof* Denote the row vectors of A by L ® £ Ax j, where x ranges over the nr r-place column vectors over GF(p?"). Then L is an L(n,k,r) space for suppose Ax, Ay, x ^ y, have as many as r components the same, say i^,ig, *.*,i‘r* Then the submatrix B of A consisting of the row vectors a. ,...,a. -“-1 1r satisfies Bx = By. But by hypothesis B is nonsingular, so x = y contradicting our choice of x and y. Hence d(Ax,Ay) > k - r, and L is an L(n,k,r) space. Finally, by Lemma 7, from A we can construct a k x (k - r) matrix with all r-rowed minors non singular, and applying the above proof we obtain an L(n,k,k - r) space. The first part of the above theorem corresponds to that of Bose and Bush [7; Th.^A, p.521 ] with index one. They employ a similar proof. THEOREM 5. For n = pm, p a prime, (1) fifyn) is r-orthogonal for k < n + 1. (2) flfyn) is3 -orthogonal for k < n + 2 and p = 2. (3) S^(n) is r-orthogonal for k < r + 1. Proof. Letting a-^,..., a^ _ ^ denote the nonzero elements of GF(pm ), one readily verifies that the matrix 2U 1 ®1 * * * **1 1 ®2 a2 1 0 • • • 0 0 0 0 • • • 0 1 has all r-rowed minors nonsingular, since their determinants all reduce to the Vandermonde type. For the special case p = 2, r * 3, we may adjoin to A as an (n + 2)-th row the vector (0,1,0) aid again verify that A has all 3 -rowed minors nonsingular. Conclusions (l) and (2) now follow from theorem h and lemma h* Finally, consider (3). For k = r, the result is trivial. For k = r + 1, the matrix obtained by adjoining the row vector (1,1,...,l) to the r x r identity matrix has all r-rowed minors nonsingular, and (3) follows from theorem I4. Conclusions (l) and (2) of the above theorem correspond to the theorem of Bush [8, p. ii313, who obtains the construction by employing polynomials over GF(pm ). From lemma U, corollary 2 of theorem 3, and theorem 5> we obtain immediately 25 THEOREM 6. If n ** n p^ 1 is the decomposition of n into distinct prime powers, then (1) ^ ( n ) is r-orthogonal for k = min (p^e^ + 1). (2) S^n) is 3-orthogonal for k = 2® + 2 if min (p-j8* + 1) * 2® + 1. (3) flfyn) is r-orthogonal for arbitrary n whenever k < r + 1. From the relation between orthogonal Latin squares and L(n,k, 2) spaces, for r ■ 2 we obtain the theorem proved in Mann's book [19, Th. 8.8, p. lo5]: OOROLLART. There exist at least min (p^e^ - 1)orthogonal Latin squares of side n = n P-j6-*-. THEOREM 7. Let L be an L(n,k,r) space, r > n - 1, k > r + 2. Then r » n - 1, and for every x in SK (n), d(x,L) < k - r. Proof. Let x « (x-^, ...,2^), and applying lemma 2 with t = r, let y = (a^, ...ja^) be the unique element of L having ai = x^, i =* 1, ...,r. Let y* *» (a^, ...,5^), and in Sr (n) consider the unit closed sphere c(y',l). The sphere contains r(n - l) elements y£, different from y", and by the triangle inequality, d(y£,y^) < 2 , i / j. Again by lemma 2, to each y£ = (b^j,...,b^r) there corresponds a unique element yi in L, yA «= (b^, ...,bir,ui,vi,...). Now d(y,y±) > k - r and d(yA,y£) - 1 imply u± / a ^ , v± / a ^ * for 1 t d(y^,y^) > k - r and d(y£,yp < 2 imply that the ordered pairs 26 (ui^i) and (tij,Vj) are distinct. Thus there are r(n - l) distinct ordered pairs (^Yji) for which tii / a r + 1 o r v i j^ a^, + 2* But since the total number of such pairs is (n - l)^, we must have r(n - l) < (n - l)^ or r < n - 1. Hence r = n - 1. We We now prove the second part of the theorem. If for either i = r + 1 or r + 2, we are done. Soassume £ a^, i = r + 1, r + 2. But then(x . x^ 0) must be one of the r + j. r + c. pairs (u£,v±), aid the conclusion follows. As an immediate corollary, we obtain the equivalent of the theorem of Bush [8, p. U27]: COROLLARY. If S^n) is r-orthogonal, then r > n implies k < r + 1. (Thus there are no Greco-Latin squares of order 2, no Greco-Latin cubes of order 3, etc.) From Tarry's result on the impossibility of Greco-Latin squares of order 6, lemmas 3,U,£ and theorems £-(2), 6-(3)> 7, it follows that there are no existence or non-existence theorems for L(n,k,r) spaces which hold for all n, beyond those already given. Therefore we now turn our attention to particular values of n. THEOREM 8. If S^(n) is r-orthogonal for r > 2 and k = n + r - 1, then ( ” J t - mod(t - 1) for t *» 2,3, ...r. - 2 3)~=° Proof. The proof is by induction on r. For r = 2 the theorem is trivial. Assume the theorem holds for r - 1, r > 3. 27 Let L be an L(n,k,r) space, and let x » (ap...^^) be in L, k ® n + r - l. By lemma 2, L contains nr"^ elements y^, j = 1,.. .,n , with first and second components and bg, respectively, b^ f a p b2 / ag. Also, for every set i p ...,ir_2 of r - 2 distinct integers from among 3,U, ..«,n + r - 1, there is a unique element y ( i p .. ‘ip-g) among the 7-j *s with i^.-th component equal to , j*l, ...,r-2. But then by lemma 6, the distance of this element from x is n, and so among the last n + r - 3 components, y(ip ...,ir_2) has r - 1 components equal to the corresponding components of x. Hence there are ” gJ “ r " 1 distinct sets ^ j p •. *,^-2^ associated with the same element y(ip ...ji^g)* Further, since x and y ( i p .. .,ir_g) agree in at most r - 1 corresponding components, there can be no more than r - 1 such sets associated with sets are divided into classes of r - 1 sets each, and we must have 0 mod (r - l). Applying the induction hypothesis and lemma 3, we obtain the theorem. From t = 3 in the above, one obtains the theorem of Bush (6, p. Ij30)s COROLLARY. For n odd, r > 3, an L(n,k,r) space satisfies k < n + r - 2. 28 Relative to our previous remarks (section 5, ex. a)), from the above theorem and theorem 5> it follows that complete sets of orthogonal Latin cubes always exist for n a power of 2, and never exist for n odd. However, for naan odd prime power > 5, we can always construct a complete set less one. THEOREM 9. If S^ n ) is 2-orthogonal, then so is S11 + ^ (n). Proof. Let L be an L(n,n,2) space in Sn(n), where S(n) - i * L— ,ah | * Deno'l:ie ky the n elements of L having first component equal to a^ (lemma 2). Let x£ in 3n + ^(n) be the element obtained by adjoining a^ as an (n + l)-th component to x^, i = 1, ...,n. Let y in L be different from the x^'s. Then since d(x^,Xj) = n - 1 for all i ^ j, each a^ occurs exactly once as a |fc-th component of the x^'s for t = 2, ...,n. Hence d(xj_,y) > n - 2 for i = 1, ...,n implies that there is a unique a. ^ ^ subscript m for which d(x^,y) = n. Let j' in S (n) be the element obtained by adjoining as an (n + l)-th component to y. Now repeat the above process for all y in L different from the x^’s, and denote by L' the set of n^ elements of ,Sn + ^"(n) which are thus obtained. Ey construction it is clear that any element of L' has distance n from each of the x£*s. Let y£ and y£ any two distinct elements of L' different from the x£’s. If d(y1,y2) » n, then d(y',y') > n. If ^ and yg have the same first component b ^ a^, let y^,...,yn denote the remaining elements of L with first component b. Then applying the argument 2? used above to the y^, for each x^ there is a unique subscriptm for which dCy^x^) =» n. Hence, by construction, no two of the y£*s have the same (n + l)-th component, and so in particular, dCy^y^) “ n * Finally, if y^ and yg have the same j-th component b, 2 < j < n, let 7 y » ) 7 n denote the remaining elements of L with j-th component b. Again applying the above argument, we obtain d(y£,y£) = n. Hence d(x',y') > n for all x',y' in L', and so L' is an L(n,n + 1,2) space and SP + ^(n) is 2-orthogonal. The above theorem is equivalent to saying that eveiy set of n - 2 orthogonal Latin squares of side n may be completed to a full set of n - 1 orthogonal Latin squares. From our remarks following theorem 8, it is interesting to note that the corresponding theorem for cubes is false. From lemmas 3 and U, theorem 9, the Bruck-Hyser nonexistence theorem [lO], and the relations among orthogonal Latin squares, projective planes, and L(n,k,r) spaces, we obtain immediately: THEOREM 10. If n = 1 or 2 (mod U) and the squarefree part of n is divisible by a prime of the form ijk + 3, then S^(n) is not r-orthogonal for k > n + r - 2, r > 2. THEOREM 11. If S^(n) is r-orthogonal, then it admits of a partitioning into pairwise disjoint, superimposable L(n,k,r) spaces. 30 Proof. Suppose S^(n) is r-orthogonal, and let L denote an L(n,k,r) space of ,Sk(n). Let A^, ...jA^ _ r he k - r Latin squares of side n, and denote by a(i,j) the permutation j 1 2 ... n 1 vhere ( a ^ a - , ...,a ) is the i-th row vector V 3! *2 *** an' n of Aj. Finally, let as ■ eo(i^, •••,ik _ r) denote the mapping of Sk(n) into itself generated by performing the permutation u(ij,j) upon the j-th components of Sk(n), j * 1, ...,k - r. It is d e a r that to is indeed a motion (definition 6), and so under to, L is carried into a superimposable L(n,k,r) space, L(i^, ...jijj _ r). Further, from lemma 2 with t = r, it is easy to see that if ( i p . . . , ^ _ ) and (j-j, ...,jk _ ) are distinct as vectors, then L(i^, _ r) and L( j^, • ••,3k _ p) are disjoint. Finally, since L(ip...,ik _ ) consists of nr elements, and there are nk "" r distinct vectors (i-^, r ), it is d e a r that these spaces exhaust S^(n). THEOREM 12. For r > 1, L(n,k,r) is a metric basis for Sk (n). Proof. The theorem is trivial for k a r, so assume k > r. Let x and y be^ arbitrary in Sk (n), x a (a^, ...,ak), y * (bj, ...,1^), x / y. The theorem will be proved if it can be shown that (C): there exists z in L(n,k,r) such that d(x,z) < d(y,z). Proof is by induction on r. Consider first L(n,k,2). Suppose, say, a^ ^ bi • Let z ,...,z be the n elements of L(n,k,2) with first 1 1 I n component equal to a^. If for some z^, d(y,z^) = k, we are done since d(x,zi) < k - 1. So suppose d(y,z^) < k for i = 1,...,n. 31 Then since L(n,k, 2) has minor diameter > k - 1, we must have d(y>Zi) “ k - 1, i * 1, ...,n, and indeed, in this event k = n + 1. Now let z^. be the unique element among the z^ (lemma 2) with second component equal to ag» Then d(y, z^) = k - 1 and d( d(x,z^) < k - 2. This completes the proof for r = 2. Now suppose that (C) holds for r -1 (r > 3) and all k, and consider L(n,k,r). If d(x,y) < r, then by lemma 2, we can always find a z in L(n,k,r) such that d(x, z) < d(y, z). So suppose y* «• T d(x,y) > r > 3. Select the n elements in L(n,k,r) with k-th component equal to a^. Denote these elements by z^, . . z^., T* — T_ t = n , and let z£, ...,z^ be the corresponding elements of S* ” 1(n) obtained by dropping the k-th components of the z^. §y the proof of lemma 3, the z£ constitute an L(n,k - l,r - l) space, and by the induction hypothesis there exists z£ in L(n,k - l,r - l) with d(x",z£) < d(y',z^), where x ** (a^,«*«,a^ _ y a But d(x',z') « d(x,zi), and d(y',z') < dfoz.^). Hence d(x,z±) < d(y,z±). 7. Future Research. The investigation of the metric properties of S^(n) and, in general, of power-sets has, of course, only its beginnings in the present paper. Indeed, one of the central problems is first of all to discover "significant" concepts (such as "extent" appears to be, for example) to study, since many of the classical metric concepts apparently will have 32 limited value, and topological concepts become completely trivial for the finite spaces. Certainly high on the list of desiderata would be a development of the basic theory to the point where the elements, say, of S^(n) could be treated abstractly, making it unnecessary to deal with their internal structure each time a new result is under scrutiny. For it is precisely at the point where internal combinatorial structure becomes too complex for the mind to grasp as a totality that our efforts fail. An aid in this direction would be a penetrating abstract characterization of S^(n). A line of attack which has been wholly neglected in the present paper and which may well prove to be the most fruitful, is an examination of the distance matrix. One may readily obtain an indication of the manner in which some of the properties of S^(n) are reflected in its distance matrix A by going through the definitions and theorems and rephrasing them in terms of A. Of course, one of the critical questions in this regard is whether these and other significant properties of A lend them selves to matric methods and theoiy. In addition to S^(n) as an object of intrinsic interest in its own right, the results thus far obtained offer hope that this type of approach may provide a useful common orientation for a wide class of combinatorial problems. Bibliography 1. Reinhold Baer, Nets and groups, Trans. Amer. Math. Soc., vol. U6, 1939, pp. 110-lljl. 2. W. W. Rouse Ball, Mathematical Recreations and Essays, Macmillan Co., New York, 19U7. 3. Grace E. Bates, Free loops and nets and their general izations, Amer. Jour, Math., vol. 69, 19U7, pp. U99-550. U. Leonard M. Blumenthal, Distance Geometries, The University of Missouri Studies, vol. 13, no.2, 1938. 5. Leonard M. Blumenthal, Theory and Applications of Distance Geometry, Oxford Univ. Press, New York, 1953. 6. R. G. Bose, On the application of the properties of Galois fields to the problem of construction of hyper-Graeco- latin squares, Sankhya, vol.3, 1938, pp. 323-338. 7. R. C. Bose and K. A. Bush, Orthogonal arrays of strength two and three, Ann. Math. Stat., vol. 23, 1952, pp. 508-52U. 8. K. A. Bush, Orthogonal arrays of index unity, Ann. Math. Stat. vol. 23, 1952, pp. U26-li3U. 9. R. H. Bruek, Finite nets, I. Numerical invariants, Can. Jour. Math, vol., 3, 195l, pp. 9U-107. 10. R. H. Bruck and H. J. Ryser, The nonexistence of certain finite projective planes, Can. Jour. Math., vol. 1, 19h9, pp. 88-93. 11. S. Ghowla and H. J. Ryser, Combinatorial problems, Can. Jour. Math., vol. 2, 1950, pp. 93-99. 12. Henry Ernest Dudeney, Amusements in Mathematics, Nelson and Sons, 1917. 13. Leonard Euler, Recherches sur une nouvelle espece de quarres magiques, Commentationes Algebraicae, Opera Omnia, series prima, vol. 7, 1923. 1U. Marshall Hall, Jr., Projective Planes, Trans. Amer. Math. Soc., vol. 5U, 19U3, pp. 229-277. 33 3U 15. M. G. Kendall, ¥ho discovered the latin square?, Amer. Stat., vol. 2, 19U8, p. 13. 16. F. ¥. Levi, Finite geometrical systems, Univ. of Calcutta, 19U2. 17. H. F. MacNeish, Euler squares, Ann. Math., vol. 23, 1921, 221-227. 18. H. B. Mann, On orthogonal latin squares, Bull. Amer. Math. Soc., vol. 50, 19UU, pp. 2it9-2f>7. 19. H. B. Mann, Analysis and Design of Experiments, Dover, New York, 19U9. 20. H. ¥. Norton, The 7 x 7 squares, Ann. I&igen., vol. 9, 1939, 269-307. 21. R. E. A. C. Paley, On orthogonal matrices, Jour. Math and Fhys., vol. 12, 1933, pp. 311-320. 22. C. R. Rao, Hypercubes of strength d leading to confounded designs in factorial experiments, Bull. Calcutta Math. Soc., vol. 38, 19U6, pp. 67-78. 23. H. J. Ity'ser, Geometries and incidence matrices, Amer. Math. Mo., vol. 62, 1955, pp. 25-31. 2l*. Esther Seiden, On the problem of construction of orthogonal arrays, Ann. Math. Stat., vol. 25, 195U, pp. 151-156. 25. ¥. L. Stevens, The completely orthogonalized latin square, Ann. Eugen., vol. 9, 1939, pp. 82-93. 26. G. Tarry, Le probleme de 36 officiers, Mathesis, vol. 20, 1901. 27. F. Yates, Incomplete randomized blocks, Ann. Eugen., vol. 7, 1936, pp. 121-lliO. Autobiography I, Robert Silverman, was b o m in Cleveland, Ohio, October 23, 1928. I received my secondary school education in the public schools of Cleveland Heights, Ohio. All of my university training has been at Ohio State University, which granted me the Bachelor of Science Degree in 1951 and the Master of Arts Degree in 195U. From 1952 to 1955, I served as an Assistant Instructor in the Statistics Laboratory of the Mathematics Department under the direction of Professor D. Ransom Whitney. In 1955 I was awarded a National Science Foundation Fellowship, and during the years 1956 and 1957 I taught mathematics Tinder a graduate assistant ship. For the past six months, while completing the requirements for the degree Doctor of Philosophy, I have been a Research Assistant on an Office of Ordnance Research project directed by Professors Marshall Hall, Jr., Erwin KLeinfeld, and Herbert J. Ryser, and devoted to a study of projective planes and related problems in combinatorial analysis. 35