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4 Orthogonal Quasigroups, Qroupoids and Triple Systems

# 4 Orthogonal Quasigroups, Qroupoids and Triple Systems

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184 Chapter 5

Corollary. Every quasigroup possesses an orthogonal complement (which is

not necessarily a quasigroup).

Proof. If (G, ·) is the given quasigroup, we define a groupoid (G, ∗) by x ∗ y =

x · xy. Then (G, ∗) is orthogonal to (G, ·) by the first part of the proof of the

theorem.

Theorem 5.4.2 A quasigroup (G, ·) which satisfies the constraint y · yx =

z · zx ⇒ y = z possesses an orthogonal complement which is a quasigroup.

Proof. Define a groupoid (G, ∗) by x∗y = x·xy. Then the equation x∗y = x∗z

is equivalent to x · xy = x · xz and implies y = z since (G, ·) is a quasigroup. The

equation y ∗ x = z ∗ x is equivalent to y · yx = z · zx and implies y = z by the

hypothesis of the theorem. It follows that (G, ∗) is a quasigroup and, by the first

part of the proof of Theorem 5.4.1, it is orthogonal to (G, ·).

Corollary. A quasigroup (G, ·) which satisfies the constraint y · yx = xy has

an orthogonal complement which is a quasigroup.

Proof. Such a quasigroup clearly satisfies the condition y · yx = z · zx ⇒ y =

z.

Let L be a latin square with transpose LT and let (G, ·) and (G, ∗) be the

quasigroups whose Cayley tables are obtained by bordering L and LT with the

first row and column of L. Then, by Theorem 5.4.2, (G, ·) and (G, ∗) are orthogonal quasigroups if x ∗ y = yx = x · xy. So, when the quasigroup (G, ·) satisfies the

identity yx = x · xy, the latin square L will be orthogonal to its own transpose.

This identity was called the Stein identity by Belousov (see Section 2.1) and he

called a quasigroup which satisfies it a Stein quasigroup [see Belousov(1965), page

102] while Sade(1960a) called such a quasigroup anti-abelian. As Stein pointed

out, quasigroups which satisfy this identity are idempotent (we have x · xx = xx

and so, by left cancellation, xx = x) and distinct elements do not commute. We

discuss such quasigroups and their associated latin squares in the next section.

Stein hoped that the above theorems might enable him to construct counterexamples to the Euler conjecture about orthogonal latin squares. However, in

this connection he was only able to prove the following:

Theorem 5.4.3 For all orders n ≡ 0, 1 or 3 (mod 4) there exist quasigroups

(G, ·) satisfying the constraint y · yx = z · zx ⇒ y = z.

Proof. Let us denote by A[GF (2k ), α, β] the groupoid constructed from the

Galois field GF (2k ) by the multiplication x ⊗ y = αx + βy, where α and β are

fixed elements of GF (2k ).

If αβ(1 + β) = 0 and k ≥ 2, the groupoid A[GF (2k ), α, β] satisfies the constraint and is a quasigroup because x ⊗ y = x ⊗ z implies αx + βy = αx + βz,

whence y = z if β = 0, and y ⊗x = z ⊗x implies αy +βx = αz +βx, whence y = z

The concept of orthogonality 185

if α = 0. As regards satisfaction of the constraint, we have x⊗(x⊗z) = y ⊗(y ⊗z)

implies αx + β(αx + βz) = αy + β(αy + βz). That is, α(1 + β)(x − y) = 0, giving

x = y if α(1 + β) = 0. The order of this quasigroup is 2k , which is congruent to

zero modulo 4 if k ≥ 2.

By forming the direct product of a system of this type with an abelian group

of odd order, it is possible to construct systems satisfying the constraint whose

orders n are congruent to 1 or 3 modulo 4. (In a group of odd order, yy = zz

implies y = z, as was proved in Theorem 1.5.3, so y · yx = z · zx implies y = z.)

The elements of the direct product will be ordered pairs (bi , ci ), bi ∈ GF (2k ),

ci ∈ (H, ·) where (H, ·) is a group of odd order. The law of composition (∗) will

be defined by (bi , ci ) ∗ (bj , cj ) = (bi ⊗ bj , ci cj ).

Stein gave the following examples of anti-abelian quasigroups.

(1) The quasigroup of order 4 with multiplication table as shown in Figure 5.4.1.

1

2

3

4

1

1

4

2

3

2

3

2

4

1

3

4

1

3

2

4

2

3

1

4

Fig. 5.4.1.

k

(2) The groupoid A[GF (pk ), α, β] when 5(p −1)/2 ≡ 1 mod pk .

(3) The groupoid A(Cn , p, q) constructed from the cyclic group Cn of order n

by the multiplication x ⊗ y = xp y q , where p and q are fixed integers, in the

case when p = q 2 , (2q + 1)2 ≡ 5 mod n and n is odd. (Such systems exist

only when 5 is a quadratic residue of n.)

The following interesting result was proved by Belousov and Gvaramiya(1966):

namely that, if a Stein quasigroup is isotopic to a group G, then the commutator

subgroup of G is in the centre of G.

In a paper published several years later than that discussed in the preceding

pages, S.K.Stein(1965) himself showed (i) that if a quasigroup (Q, ·) of order

n has a transitive set of n automorphisms, then there is a quasigroup (Q, ∗)

orthogonal to it (cf. Theorem 7.2.1) and (ii) that no quasigroup of order 4k+2 has

a transitive automorphism group. It follows from (ii) that no counter-examples

to the Euler conjecture can be constructed by means of (i).

The properties of systems of orthogonal quasigroups have been investigated

by Belousov(1962). Also, Sade(1958c) obtained a number of interesting and useful results concerning orthogonal groupoids, their parastrophes (conjugates) and

their isotopes. In a later paper [Sade(1960a)], which we shall discuss in Section 11.2, the same author achieved the goal of Stein: namely, to construct

counter-examples to the Euler conjecture. He used a particular type of direct

186 Chapter 5

0

1

L1 = 2

3

4

1

2

3

4

0

2

3

4

0

1

3

4

0

1

2

4

0

1

2

3

0

4

3

2

1

1

0

4

3

2

2

1

0

4

3

3

2

1

0

4

4

3

2

1

0

0

2

L2 = 4

1

3

1

3

0

2

4

2

4

1

3

0

3

0

2

4

1

4

1

3

0

2

0

3

1

4

2

1

4

2

0

3

2

0

3

1

4

3

1

4

2

0

4

2

0

3

1

0

3

L3 = 1

4

2

1

4

2

0

3

2

0

3

1

4

3

1

4

2

0

4

2

0

3

1

0

2

4

1

3

1

3

0

2

4

2

4

1

3

0

3

0

2

4

1

4

1

3

0

2

0

4

L4 = 3

2

1

1

0

4

3

2

2

1

0

4

3

3

2

1

0

4

4

3

2

1

0

0

1

2

3

4

1

2

3

4

0

2

3

4

0

1

3

4

0

1

2

4

0

1

2

3

0

4

3

2

1

0

0

4

3

2

1

4

1

0

4

3

2

3

2

1

0

4

3

2

3

2

1

0

4

1

4

3

2

1

0

0

4

3

2

1

0

0

3

1

4

2

4

1

4

2

0

3

3

2

0

3

1

4

2

3

1

4

2

0

1

4

2

0

3

1

0

4

3

2

1

0

0

1

2

3

4

4

3

4

0

1

2

3

1

2

3

4

0

2

4

0

1

2

3

1

2

3

4

0

1

0

4

3

2

1

0

0

2

4

1

3

4

2

4

1

3

0

3

4

1

3

0

2

2

1

3

0

2

4

1

3

0

2

4

1

Fig. 5.4.2.

product of quasigroups which he called “produit direct singulier” and which preserves orthogonality.

The idea of forming new quasigroups as singular direct products came to Sade

as a consequence of his investigations of the related concept of singular divisors

of quasigroups introduced by him earlier in Sade(1950) and discussed further in

Sade(1953a,1957). In the opinion of the authors, this concept is a fruitful one

but did not at the time receive the attention it deserved.

We end this discussion of orthogonal quasigroups with a theorem of Barra(1963)

which we shall use on several occasions later in connection with constructions of

pairs of orthogonal latin squares.

Theorem 5.4.4 From a given set of t ≤ n − 1 mutually orthogonal quasigroups

of order n, a set of t mutually orthogonal quasigroups of the same order n (but

usually different from those of the original set) can be constructed of which t − 1

are idempotent quasigroups.

The concept of orthogonality 187

Proof. Let the multiplication tables of the quasigroups be given by the mutually orthogonal latin squares L1 , L2 , . . . , Lt all of which are bordered in the same

way. We first rearrange the rows of all the latin squares simultaneously in such

a way that the entries of the leading diagonal of one square, say the square L1 ,

are all equal. Since the rearranged squares L2 , L3 , . . . , Lt are all orthogonal to

L1 , the entries of the leading diagonal of each of them must form a transversal:

that is, be all different. A relabelling of all the entries in any one of the squares

does not affect its orthogonality to the remainder, so we may suppose that the

squares L3 , L4 , . . . , Lt are relabelled in such a way that the entries of the leading

diagonal of each become the same as those of L2 . If now each of the squares

is bordered by its elements in the order in which these elements appear in the

leading diagonal of L2 , the resulting Cayley tables define mutually orthogonal

quasigroups Q1 , Q2 , . . . , Qt of which all but the first are idempotent.

An example of the process described in the proof of Theorem 5.4.4 is given

in Figure 5.4.2.

A similar theorem was proved by N.S.Mendelsohn(1971c) who pointed out

that, when n is a prime power, it is always possible to construct sets of n − 2

mutually orthogonal idempotent latin squares.

We mention here a curiousity: In Norton and Stein(1956), with each idempotent latin square of order n an integer N associated with that square has been

introduced whose value is dependent on the disposition of the off-diagonal elements of the square (that is, those not on the main left-to-right diagonal). It has

been proved that the relation N ≡ n(n − 1)/2 modulo 2 always holds whatever

the explicit value of N .

Orthogonal triple systems

We showed in Section 2.3 that Steiner and Mendelsohn triple systems can

both be represented as quasigroups. If the quasigroups which correspond to two

triple systems of the same kind and size are orthogonal [or, more correctly, perpendicular in the case of Steiner triple systems (see later)], we say that the triple

systems are orthogonal. In the case of Steiner triple systems, the quasigroups are

totally symmetric and idempotent and so an equivalent statement is that two

Steiner triple systems of the same order and defined on the same set of elements

are orthogonal if (i) the systems have no triples in common and if (ii) when two

pairs of elements appear with the same third element in one system, then they

appear with distinct third elements in the other system. As an illustration, we

give the following two orthogonal Steiner triple systems of order 7:

S1 = (1 2 6), (2 3 7), (3 4 1), (4 5 2), (5 6 3), (6 7 4), (7 1 5)

S2 = (1 2 4), (2 3 5), (3 4 6), (4 5 7), (5 6 1), (6 7 2), (7 1 3)

In the case of Mendelsohn triple systems, the corresponding quasigroups are

semi-symmetric and idempotent: that is, each satisfies the identities x(yx) =

y and xx = x. It turns out that the conditions for two such systems to be

188 Chapter 5

orthogonal are the same as those for Steiner triple systems. We remind the

reader that we showed in Section 2.1 that x(yx) = y ⇔ (xy)x = y.

At the time of writing, the question of whether and when directed triple

systems are orthogonal had not been considered. We recall from Section 2.3 that

when such a system can be represented by a quasigroup, it is called a latin

directed triple system.

In O’Shaughnessy(1968), that author gave a construction for Room squares12

which employed a pair of orthogonal Steiner triple systems. For this purpose, he

successfully constructed orthogonal pairs of orders 7, 13 and 19 but was not able

to construct a pair of order 9. This led him to conjecture that orthogonal pairs

exist of all orders v ≡ 1 mod 6 but that they do not exist for orders v ≡ 3 mod

6. Mullin and N´emeth(1969b) proved that orthogonal pairs exist when v ≡ 1

mod 6 is a prime power and in Mullin and N´emeth(1970a) that they do not

exist when v = 9. See also N.S.Mendelsohn(1970) for some earlier results. Then

Rosa(1974) constructed an orthogonal pair of order v = 27, thus disproving the

O’Shaughnessy conjecture. After many partially successful attempts to resolve

the existence question, it was finally proved in a joint paper by Colbourne, Gibbons, Mathon, Mullin and Rosa(1994) that orthogonal pairs of Steiner triple

systems exist for all v ≡ 1 or 3 mod 6 and v ≥ 7 except v = 9. The method used

by these authors in their construction requires a generalization of the notion

of orthogonality to group divisible designs and also the concept of parastrophic

orthogonal quasigroups which we introduce in the next section.

As regards the corresponding question for pairs of Mendelsohn triple systems,

it was shown by Bennett and Zhu(1992), who again made use of parastrophic orthogonal quasigroups, that self-orthogonal (which we define below) Mendelsohn

triple systems exist of all orders v ≡ 1 mod 3 except v = 10. Later, in Bennett,

Zhang and Zhu(1996), those authors showed that such systems also exist of all

orders v ≥ 15 and v ≡ 0 mod 3 except possibly v = 18. This answers the question

of existence in the affirmative for all v ≡ 0 or 1 except v = 3, 6, 10 (which had

earlier been ruled out) and possibly v = 12 and 18.

5.5

Self-orthogonal and other parastrophic orthogonal latin squares

and quasigroups

We showed in Section 1.4 that every latin square is associated with six parastrophes (also called conjugates) including itself of which 1, 2, 3 or all 6 may be

distinct. For particular latin squares, it may happen that some or all of these

parstrophes are orthogonal. A considerable number of papers have been published which investigate this situation. In particular, every latin square which is

not symmetric has a transpose distinct from itself. A latin square which is orthogonal to its own transpose has been mis-called self-orthogonal. It is believed

12 We

define Room squares and explain O’Shaughnessy’s construction in Section 6.4.

The concept of orthogonality 189

that N´emeth was the first to use this name in his Ph.D. thesis. Shortly afterwards, the name was adopted by N.S.Mendelsohn(1969,1971b). The name has

The first to write about self-orthogonal latin squares were S.K.Stein(1957)

topic of anti-abelian quasigroups, pointing out that an anti-abelian quasigroup

has a multiplication table whose body is a self-orthogonal latin square. We have

already described some of their results in the preceding section. Both authors

and later also Mendelsohn(1971b) obtained results equivalent to the following:

Theorem 5.5.1 If a, b, a + b and a − b are integers prime to h in the ring Zh of

residue classes modulo h, then the law of composition x · y = ax + by + c defines

an anti-abelian quasigroup (Zh , ·) on the set Z.

Proof. The quasigroup (Zh , ·) will be orthogonal to its own transpose provided

that, for all x, y, z, t, xy = zt ⇒ yx = tz. So suppose on the contrary that

xy = zt and yx = tz. These conditions become ax + by ≡ az + bt (mod h)

and ay + bx ≡ at + bz (mod h). Therefore, a(x − z) ≡ b(t − y) (mod h) and

b(x − z) ≡ a(t − y). From the first equality, ab(x − z) ≡ b2 (t − y) and, from the

second, ab(x − z) ≡ a2 (t − y). Thence, (a − b)(a + b)(t − y) ≡ 0 (mod h), and so

y = t whence also x = z.

Since a, b, a + b and a − b are all to be prime to h, we find that h has to be

prime to 6 since, if a and b are both odd, then a + b is even and so h must be

prime to 2; if neither a nor b is a multiple of 3 then either a + b or a − b is a

multiple of 3 whence h must also be prime to 3. For all such integers h, antiabeian quasigroups and consequently pairs of orthogonal latin squares of order

h such that one is the transpose of the other, actually exist. Sade takes as an

example, a = 2 and b = −1, from which we see that the operation x · y = 2x − y

gives an anti-abeian (and idempotent) quasigroup whenever h is an integer prime

to 6.

Theorem 5.5.2 In any Galois field GF (pn ), every quasigroup defined on the

elements of the field by a law of composition of the form x · y = ax + by + c,

where a and b are non-zero elements of the field such that a2 − b2 = 0, is antiabelian. Moreover, such quasigroups exist in all finite fields except GF (2) and

GF (3).

Proof. Exactly as in Theorem 5.5.1, the conditions xy = zt and yx = tz

together imply (a2 − b2 )(t − y) = 0 if a and b are non-zero. Then, provided that

a2 − b2 = 0, we have y = t and x = z, so xy = zt ⇒ yx = tz as required for an

anti-abelian quasigroup. In any field except GF (2) and GF (3) non-zero elements

a and b with distinct squares exist. For example, we can take a = 1 and b any

element of the field distinct from 0, 1, −1.

In the early 1970s, several papers were devoted to providing constructions for

self-orthogonal latin squares of order n for isolated values of n and for various

190 Chapter 5

infinite classes of n. These are listed in the bibliography of Brayton, Coppersmith

and Hoffman(1976). The problems of existence and construction were brought

to the attention of the latter authors by a question concerning a particular type

of mixed-doubles tennis tournament (which was christened “spouse-avoiding”)

because these authors discovered that such tournaments involving n married

couples can be constructed when and only when a self-orthogonal latin square

of order n exists. Full details can be found in the aforementioned paper, in a

summary paper of Brayton, Coppersmith and Hoffman(1974), and also in Keedwell(2000). Brayton et al were able to prove that self-orthogonal latin squares

exist of all orders n except n = 2, 3 and 6.

In T.Evans(1973), that author considered a generalization of self-orthogonality

to latin cubes (which latter we define later in this chapter).

Next, the more general question of existence of latin squares which are orthogonal to any one or more of their parastrophes was considered. Phelps(1978)

showed that, if there exists a latin square of order n which is orthogonal to its

(2, 3, 1)-parastrophe, then we can construct from it a latin square of the same

order which is orthogonal to its (3, 1, 2)-parastrophe and conversely. An exactly

similar statement can be made about the (3, 2, 1)- and (1, 3, 2)-parastrophes.

Phelps proved existence of, and gave constructions for, latin squares orthogonal

to their (2, 3, 1)-parastrophe for all orders n except 2 and 6. He also obtained

a similar result for latin squares orthogonal to their (3, 2, 1)-parastrophe except

that, in the latter case, he was not able to guarantee existence for the orders

n = 14 and 26.

Belousov(1983b,2005) and, independently, Bennett and Zhu(1992) considered

the related question of which quasigroup identities would ensure that the quasigroups defined by such identities would be orthogonal to one or more of their

parastrophes. This latter question had earlier been considered by T.Evans(1975)

and by Lindner and N.S.Mendelsohn(1973).

Belousov began by considering identities in an algebra Q(Σ), where Σ is

some system of quasigroup operations (quasigroups) defined on a set Q (cf. Section 2.2). He showed that a non-trivial quasigroup identity must be of minimum

length five and must involve two different free variables, one appearing twice and

the other three times (as for example in the Stein identity x·xy = yx). It is convenient for the following discussion if we use A, B, C, etc. to denote quasigroup operations so that, for example, we write A(x, B(x, y)) = y instead of x ·(x◦ y) = y,

where A, B are respectively the binary operations (·) and (◦) operating on a set

Q of elements.13 Belousov proved that any non-trivial minimal identity defined

in (Q, Σ) can be written in the form A(x, B(x, C(x, y))) = y, where A, B, C represent three operations possibly all different. For example, he showed that the

identity A(B(x, y), C(x, y)) = x can be re-written as A(13) (x, C(x, y)) = B(x, y)

and thence as A(13) (x, C(x, B (23) (x, z))) = z, where z = B(x, y). He defined

a special binary operation E (not a quasigroup operation) by E(x, y) = y.

13 See

Section 2.1 for an earlier discussion of notation for parastrophes.

The concept of orthogonality 191

Then, for brevity, he abbreviated the minimal identity A(x, B(x, C(x, y))) = y to

ABC = E and remarked that it is easy to see that a quasigroup which satisfies

this minimal identity also satisfies the minimal identities BCA = E, CAB = E,

C r B r Ar = E, B r Ar C r = E and Ar C r B r = E, where r denotes the permutation

(2 3). Next, he proved

Belousov Lemma. Let A, B be quasigroup operations. Then A, B are orthogonal

operations if and only if there is a quasigroup operation K such that

K(x, B(x, (A(23) (x, y))) = E(x, y).

T.Evans(1975) had earlier and independently proved an equivalent lemma for

the special case of parastrophic operations: namely,

Evans Lemma. Let A, B be parastrophic (or conjugate) operations on a quasigroup. Then A, B are orthogonal operations if and only if there is a further

operation L such that L(A(x, y), B(x, y)) = x.

Evans called an identity of the type just described a short conjugate-orthogonal

identity. Belousov, on the other hand, called a quasigroup Q(A) which satisfies

an identity of the form Aα (x, Aβ (x, Aγ (x, y))) = y, where Aα , Aβ , Aγ are parastrophic operations (that is, operations from the set P = {A, A(12) , A(13) , A(23) ,

A(123) , A(132) } as defined in Section 2.1), a Π-quasigroup of type [α, β, γ].

By writing the minimal identity ABC = E in the form ABC rr = E, it follows

directly from Belousov’s lemma that the quasigroup operation B is orthogonal

to the operation C (2 3) (which we shall write as B⊥C r ) and so also C⊥Ar and

A⊥B r .

It follows from this that, if Q(A) is of type [α, β, γ], then Aβ ⊥Aγr , Aγ ⊥Aαr

and Aα ⊥Aβr .

Since (as we remarked above) a quasigroup which satisfies the minimal identity ABC = E also satisfies other minimal identities of this canonical form, we

may expect that a Π-quasigroup Q(A) of type [α, β, γ] will also be of other types

as well. (For example, it will be of type [β, γ, α].) The types [α, β, γ)] and [β, γ, α]

are said to be parastrophically equivalent.

Belousov showed that there are just seven parastrophically inequivalent types

of minimal identity such that a quasigroup which satisfies one of these identities

is orthogonal to one or more of its parastrophes.

We list these in Figure 5.5.1 which is taken from Table 1 in Belousov(2005).

In that table, r represents the permutation (2 3) as before and l represents

the permutation (1 3). However, we treat permutations as right-hand mappings

whereas Belousov treats them as left-hand mappings, so the table below differs

from that in Belousov(2005). We give identities (3) and (6) from the table as

examples.

Consider the identity No. 3. x(x(y/x)) = y.

Let y/x = z. Then y = zx. So A(z, x) = y, whence A(13) (y, x) = z.

Then A(13)(12) (x, y) = z.

192 Chapter 5

No.

Type

Identity

Derived form

1.

T1 = [1, 1, 1]

x(x · xy) = y

x(x · xy) = y

2.

T2 = [1, 1, l]

x(x(x/y)) = y

x(y · yx) = y

3.

T4 = [1, 1, rl]]

x(x(y/x)) = y

x · xy = yx

4.

T6 = [1, l, rl]

x(x/(y/x)) = y xy · x = y · xy

5.

T10 = [1, lr, l]

6.

T8 = [1, lr, rl]

7.

T11 = [1, rl, lr] x((y\x)/x) = y

(x/xy)/x = y

xy · yx = y

x((y/x)\x) = y xy · y = x · xy

yx · xy = y

Note

Stein’s 1st law

Stein’s 2nd law

Stein’s 3rd law

Schră

oders 1st law

Schră

oders 2nd law

Fig. 5.5.1.

Now, (1 3)(1 2) = (1 3 2) = (2 3)(1 3) = rl. So, Arl (x, y) = z.

Therefore, y/x = z ⇔ Arl (x, y) = z and so the above identity can be written as

A(x, A(x, Arl (x, y))) = y or as [1, 1, rl].

Consider the identity No. 6. x((y/x)\x) = y.

Let y/x = z as before [so that Arl (x, y) = z] and let v = z\x.

Then zv = x or, equivalently, A(123) (x, z) = v where the identity (6) is xv = y

or A(x, A(123) (x, z)) = y. Since (1 2 3) = (1 3)(2 3) = lr, the identity (6) is

A(x, Alr (x, Arl (x, y))) = y or [1, lr, rl].

As an example of how the table may be used, consider the Π-quasigroup

which satisfies the identity [1, 1, rl]. Since rl = (2 3)(1 3) = (2 1 3), this is the

identity

A(x, A(x, A(2 1 3) (x, y))) = y.

(2 1 3)

14

Let A

(x, y) = z. Then z · x = y so the identity becomes x · xz = zx which

is Stein’s first law. From above, we find that A⊥Arlr , Arl ⊥Ar and A⊥Ar . Thus,

in particular, Q(A) is orthogonal to its parastrophes Q(A(1 2) ) and Q(A(2 3) ).

The first of these implies that Q(A) is self-orthogonal.

A full list of orthogonalities between parastrophic operations in Π-quasigroups

is given in Belousov’s paper.

Working independently, Bennett and Zhu(1992) obtained the same result.

However, the list of seven inequivalent quasigroup identities obtained by the latter authors differed slightly from that obtained by Belousov in that the identities

T1 and T6 were replaced by their duals and T2 was replaced by the dual of Belousov’s T5 = [1, 1, s], where s = (1 2), which is parastrophically equivalent to

T2 .

For further details of this topic, the reader should refer to the very extensive

and detailed papers of Belousov(1983b,2005) and Bennett and Zhu(1992).

14 If

A(u, v) = w, then Ar (u, w) = v and so Arl (v, w) = u. Thus, Arl (x, y) = z ⇒ A(z, x) = y.

The concept of orthogonality 193

In some more recent work on self-orthogonal latin squares by Graham and

Roberts(1991,2002), these authors have considered maximal sets of pairwise orthogonal self-orthogonal latin squares, say {A1 , AT1 , A2 , AT2 , . . . , Am , ATm } of order n. They have shown that, when n = pk , p prime, this maximal number is

(2k − 2)/2 when p = 2 and is (pk − 3)/2 otherwise. [cf. Theorem 5.1.2 and

Theorem 5.3.1.] They have given constructions for such sets using an affine

plane and/or a left nearfield. In a later paper, Graham and Roberts(2007), the

same authors have established a relationship between complete sets of orthogonal self-orthogonal latin squares and projective planes analogous to that of

Theorem 5.2.2.

5.6

Orthogonality in other structures related to latin squares

In this section we consider how the orthogonality concept may be generalized

to apply to a number of structures related to latin squares. We consider in turn

latin rectangles, permutation cubes, latin cubes and hypercubes and orthogonal

arrays.

Latin rectangles were defined in Section 3.1. We say that two latin rectangles

of the same size are orthogonal if, when one is superimposed on the other, each

ordered pair of symbols (r, s) occurs in at most one cell of the superimposed

pair. Also, a set of n − 1 mutually orthogonal latin rectangles of size m × n, with

m ≤ n is a complete set.

It is easy to see that the definition of orthogonality for latin squares is included as a special case of this more general definition and, by the method of

Theorem 5.1.2, that one cannot have more than n − 1 mutually orthogonal m × n

latin rectangles if 2 ≤ m ≤ n.

The following result was proved by Quattrocchi(1968).

Theorem 5.6.1 For every prime p and integer q such that q has no prime divisor less than p there exists at least one complete set of mutually orthogonal

p × pq latin rectangles.

Proof. Throughout this proof equivalences will be modulo pq. For each k =

1, 2, . . . , pq − 1, we define a p × pq matrix Rk = ||αij ||, i = 0, 1, . . . , p − 1;

j = 0, 1, . . . , pq − 1, by αij ≡ ik + j and 0 ≤ αij ≤ pq − 1.

It is immediate that each of the integers 0, 1, . . . , pq − 1 occurs exactly once

in each row of Rk and at most once in each column. Consequently, Rk is a p × pq

latin rectangle.

Let us consider two rectangles Rk1 and Rk2 . Suppose that when they are

placed in juxtaposition the ordered pair (s, t) appears both in the cell in row i1

and column j1 and in the cell in row i2 and column j2 . Without loss of generality

we may assume that i1 > i2 . Then by the definition of the Rk we have

i1 k1 + j1 ≡ s,

and

i1 k2 + j1 ≡ t,

194 Chapter 5

i2 k1 + j2 ≡ s,

i2 k2 + j2 ≡ t.

These relations imply that

(i1 − i2 )k1 + (j1 − j2 ) ≡ 0 ≡ (i1 − i2 )k2 + (j1 − j2 )

(5.6)

(i1 − i2 )(k1 − k2 ) ≡ 0.

(5.7)

and so

Now recall that q has no prime divisor less than p and 0 ≤ i1 < i2 ≤ p − 1. Hence

i1 − i2 is relatively prime to pq, so (5.7) implies k1 ≡ k2 which implies that we

did not start with distinct rectangles. This is sufficient to show that the pq − 1

rectangles are pairwise orthogonal.

Quattrocchi made use of this theorem in a construction of generalized affine

spaces (equivalent to a certain type of balanced incomplete block design) from

similar spaces of smaller order. The latin rectangles were used to define the incidence relation between point and line in the synthesized structure. Much later,

Mullen and Shiue(1991) used Quattrocchi’s construction to build orthogonal

latin rectangles of more general sizes than those constructed by Theorem 5.6.1.

ˇ an

Hor´ak, Rosa and Sir´

ˇ(1997) considered pairs of what they called maximal

orthogonal rectangles, which are orthogonal r × n latin rectangles which cannot

be extended to (r + 1) × n latin rectangles. They conjectured that for sufficiently

large n such pairs exist for precisely those r which satisfy n/3 < r ≤ n, and they

proved some results in that direction.

We remark here that Wanless(2001) has given a construction for four mutually orthogonal 9 × 10 latin rectangles and, moreover, these form a latin power

set as defined in Section 10.2.15

Finally, Asplund and Keranen(2011) have introduced what they call equitable

latin rectangles and have shown how to construct mutually orthogonal sets of

these.

A latin square is a two-dimensional object and the latin rectangle is a generalization of it in the sense that the “size” of one of these dimensions is allowed

to be different from the other. A different generalization is obtained if, while

retaining a fixed size, we allow the number of dimensions to be increased. If we

increase the number of dimensions to three, we obtain what should properly be

called a latin cube; an object having n2 rows, n2 columns and n2 files such that

each of a set of n elements occurs once in each row, once in each column and

once in each file.

An illustrative example for the case n = 3 is given in Figure 5.6.1. If the

number of dimensions increases still further to m say, we obtain an object which

could reasonably be called an m-dimensional latin hypercube. Unfortunately, the

terms latin cube and latin hypercube have been used by statisticians to denote

15 It is interesting to compare this result with Brouwer’s construction of four almostorthogonal 10 × 10 latin squares mentioned on pages 147, 149 and elsewhere in [DK2]. ### Tài liệu bạn tìm kiếm đã sẵn sàng tải về

4 Orthogonal Quasigroups, Qroupoids and Triple Systems

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