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
stuck and so we shall follow convention and adopt it.
The first to write about self-orthogonal latin squares were S.K.Stein(1957)
and Sade(1960a). However, both writers considered instead the closely related
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].