DOI: 10.2478/v10324-012-0014-3
Analele Universităţii de Vest,
Timişoara
Seria Matematică – Informatică
L, 2, (2012), 45– 53
On a Group of Linear-Bivariate Polynomials
that Generate Quasigroups over the Ring Zn
T. G. Jaiyéo.lá and E. Ilojide
Abstract. In this study, some linear-bivariate polynomials
P (x, y) = a + bx + cy that generate quasigroups over the ring Zn
are studied. By defining a suitable binary operation ∗ on the set
HP (Zn ) of all linear-bivariate polynomials of the form Pf (x, y) =
f1 (a, b, c) + f2 (a, b, c)x + f3 (a, b, c)y where f1 , f2 , f3 : Zn × Zn ×
Zn −→ Zn , it is proved that HP (Zn ), ∗ is a monoid. Necessary and sufficient conditions for it to be a group and abelian
group are established. If PP (Zn ) is the set of the linear-bivariate
polynomials that generate the quasigroups that are the parastrophes of thequasigroup generated
by P (x, y), then it is shown that
PP (Zn ), ∗ ≤ HP (Zn ), ∗ . The group PP (Zn ) is found to be isomorphic to the symmetric group S3 and to SPP (Zn ) ≤ S6 . A Bol
loop of order 36 is constructed using the group PP (Zn ).
AMS Subject Classification (2000). 20N02, 20NO5.
Keywords. Quasigroups, parastrophes, linear-bivariate polynomials.
1
Introduction
Let G be a non-empty set. Define a binary operation (·) on G. (G, ·) is
called a groupoid if G is closed under the binary operation (·). A groupoid
(G, ·) is called a quasigroup if the equations a·x = b and y ·c = d have unique
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T. G. Jaiyéo.lá and E. Ilojide
An. U.V.T.
solutions for x and y for all a, b, c, d ∈ G. A quasigroup (G, ·) is called a loop
if there exists a unique element e ∈ G called the identity element such that
x · e = e · x = x for all x ∈ G.
A function f : S × S → S on a finite set S of size n > 0 is said to be a
Latin square (of order n) if for any value a ∈ S both functions f (a, ·) and
f (·, a) are permutations of S. That is, a Latin square is a square matrix with
n2 entries of n different elements, none of them occurring more than once
within any row or column of the matrix.
Definition 1.1. A pair of Latin squares
f1 (·, ·) and f2 (·, ·) is said to be
orthogonal if the pairs f1 (x, y), f2 (x, y) are all distinct, as x and y vary.
For every quasigroup (G, ·), there exists five other corresponding quasigroups.
Definition 1.2. (Parastrophes)
Let (G, θ)be a quasigroup.
The five parastrophes
of (G, θ) are (G, θ∗ ), (G, θ−1 ),
(G,−1 θ), G, (θ−1 )∗ and G, (−1 θ)∗ whose binary operations θ∗ , θ−1 , −1 θ,
(θ−1 )∗ and (−1 θ)∗ defined on G satisfy the conditions :
(a) yθ∗ x = z ⇔ xθy = z ∀ x, y, z ∈ G;
(b) xθ−1 z = y ⇔ xθy = z ∀x, y, z ∈ G;,
(c) z
−1
θy = x ⇔ xθy = z ∀x, y, z ∈ G;
(d) z(θ−1 )∗ x = y ⇔ xθy = z ∀x, y, z ∈ G; and
(e) y(−1 θ)∗ z = x ⇔ xθy = z ∀ x, y, z ∈ G.
A quasigroup which is equivalent to all its parastrophes is called a totally
symmetric quasigroup while its loop is called a Steiner loop.
As remarked in Jaiyeola [6], if (L, θ) is a groupoid, then, (L, θ∗ ) is also a
groupoid while the other parastrophes are not defined. If (L, θ) is a quasigroup (loop), (L, θ∗ ) is also a quasigroup(loop) while the other adjugates are
quasigroups. Furthermore, if (L, θ) is a loop, then
(L, θ−1 ) and (L, (−1 θ)∗ )
have left identity elements, that is, they are left loops while
(L, −1 θ) and (L, (θ−1 )∗ )
have right identity elements, that is, they are right loops.
(L, θ−1 ) or (L, −1 θ) or (L, (θ−1 )∗ ) or (L, (−1 θ)∗ )
is a loop if and only if (L, θ) is a loop of exponent 2.
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The basic text books on quasigroups, loops are Pflugfelder [7], Bruck [1],
Chein, Pflugfelder and Smith [2], Dene and Keedwell [3], Goodaire, Jespers
and Milies [4], Sabinin [9], Smith [10], Jaı́yéo.lá [5] and Vasantha Kandasamy
[13].
Definition 1.3. (Bivariate Polynomial)
A bivariate polynomial is a polynomial in two variables, x and y of the form
P (x, y) = Σi,j aij xi y j .
Definition 1.4. (Bivariate Polynomial Representing a Latin Square)
A bivariate polynomial P (x, y) over Zn is said to represent (or generate) a
Latin square if (Zn , ∗) is a quasigroup where ∗ : Zn × Zn → Zn is defined
by x ∗ y = P (x, y) for all x, y ∈ Zn .
In 2001, Rivest [8] studied permutation polynomials (PPs) over the ring
(Zn , +, ·) where n is a power of 2: n = 2w . This is based on the fact that
modern computers perform computations modulo 2w efficiently (where w =
2, 8, 16, 32 or 64 is the word size of the machine), and so it was of interest to
study PPs modulo a power of 2. Below are some important results from his
work.
Theorem 1.1. (Rivest [8])
A bivariate polynomial P (x, y) = Σi,j aij xi y j represents a Latin square modulo
n = 2w , where w ≥ 2, if and only if the four univariate polynomials P (x, 0),
P (x, 1), P (0, y), and P (1, y) are all permutation polynomial modulo n.
Theorem 1.2. (Rivest [8])
There are no two polynomials P1 (x, y), P2 (x, y) modulo 2w for w ≥ 1 that
form a pair of orthogonal Latin squares.
In 2009, Vadiraja and Shankar [12] motivated by the work of Rivest continued
the study of permutation polynomials over the ring (Zn , +, ·) by studying
Latin squares represented by linear and quadratic bivariate polynomials over
Zn when n 6= 2w with the characterization of some PPs. Some of the main
results they got are stated below.
Theorem 1.3. (Vadiraja and Shankar [12])
A bivariate linear polynomial a + bx + cy represents a Latin square over Zn ,
n 6= 2w if and only if one of the following equivalent conditions is satisfied:
(i) both b and c are coprime with n;
(ii) a+bx, a+cy, (a+c)+bx and (a+b)+cy are all permutation polynomials
modulo n.
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T. G. Jaiyéo.lá and E. Ilojide
An. U.V.T.
Theorem 1.4. (Vadiraja and Shankar [12])
If P (x, y) is a bivariate polynomial having no cross term, then P (x, y) gives a
Latin square if and only if P (x, 0) and P (0, y) are permutation polynomials.
Theorem 1.5. (Vadiraja and Shankar [12])
Let n be even and P (x, y) = f (x) + g(y) + xy be a bivariate quadratic polynomial, where f (x) and g(y) are permutation polynomials modulo n. Then
P (x, y) does not give a Latin square.
The authors were able to establish the fact that Rivest’s result for a bivariate
polynomial over Zn when n = 2w is true for a linear-bivariate polynomial over
Zn when n 6= 2w . Although the result of Rivest was found not to be true
for quadratic-bivariate polynomials over Zn when n 6= 2w with the help of
counter examples, nevertheless some of such squares can be forced to be Latin
squares by deleting some equal numbers of rows and columns.
Furthermore, Vadiraja and Shankar [12] were able to find examples of pairs
of orthogonal Latin squares generated by bivariate polynomials over Zn when
n 6= 2w which was found impossible by Rivest for bivariate polynomials over
Zn when n = 2w .
Theorem 1.6. (Solarin and Sharma [11])
Let H be a subgroup of a non-abelian group G and let A = H × G. For
(h1 , g1 ), (h2 , g2 ) ∈ A, define
(h1 , g1 ) ◦ (h2 , g2 ) = (h1 h2 , h2 g1 h−1
2 g2 )
then (A, ◦) is a Bol loop.
The objective of the present study is to build a semigroup, a group and a
Bol loop using linear-bivariate polynomials over the ring Zn which generate
quasigroups.
2
Main Results
Theorem 2.1. Let P1 (x, y) = P (x, y) = a + bx + cy represent a quasigroup
over Zn such that b and c are invertible in Zn . Let Pi (x, y), i = 2, 3, 4, 5, 6
denote the linear-bivariate polynomials that represent the parastrophes of
∗
−1
−1
(G,
θ) = (G,
P1 ): (G, θ ) =(G, P2 ), (G,
θ ) = (G, P3 ), (G, θ) = (G, P5 ),
G, (θ−1 )∗ = (G, P4 ) and G, (−1 θ)∗ = (G, P6 ). Then,
(i) P2 (x, y) = a + cx + by;
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(ii) P3 (x, y) = −ac−1 − bc−1 x + c−1 y;
(iii) P4 (x, y) = −ac−1 + c−1 x − bc−1 y;
(iv) P5 (x, y) = −ab−1 + b−1 x − cb−1 y;
(v) P6 (x, y) = −ab−1 − cb−1 x + b−1 y.
Proof
(i) Let (G, θ) = (G, P1 ) be generated by P (x, y) = a + bx + cy. Now, xθy =
a + bx + cy = z ⇔ yθ∗ x = z = a + bx + cy, xθ∗ y = a + by + cx =
a + cx + by = P2 (x, y).
(ii) Let (G, θ) = (G, P1 ) be generated by P (x, y) = a + bx + cy. Now, xθy =
z ⇔ xθ−1 z = y. But a + by + cy = z ⇔ y = c−1 (z − a − bx) = xθ−1 z =
−ac−1 − bc−1 x + c−1 y = P3 (x, y).
P4 (x, y), P5 (x, y) and P6 (x, y) are established in a similar manner.
Theorem 2.2. Let P1 (x, y) = P (x, y) = a + bx + cy represent a quasigroup
over Zn and let
HP (Zn ) = {Pf (x, y) = f1 (a, b, c) + f2 (a, b, c)x + f3 (a, b, c)y}
| f1 , f2 , f3 : Zn × Zn × Zn → Zn }.
For all Pf , Pg ∈ HP (Zn ), where Pf (x, y) = f1 (a, b, c)+f2 (a, b, c)x+f3 (a, b, c)y
and Pg (x, y) = g1 (a, b, c) + g2 (a, b, c)x + g3 (a, b, c)y, define ∗ on HP (Zn ) as
follows:
Pf ∗ Pg = (Pf )g = g1 f1 (a, b, c), f2 (a, b, c), f3 (a, b, c) +
g2 f1 (a, b, c), f2 (a, b, c), f3 (a, b, c) x + g3 f1 (a, b, c), f2 (a, b, c), f3 (a, b, c) y.
(i) HP (Zn ), ∗ is a monoid.
(ii) HP (Zn ), ∗ is a group if and only if for any Pf , Pg ∈ HP (Zn ),
g1 (f1 , f2 , f3 ) = a
g2 (f1 , f2 , f3 ) = b
g3 (f1 , f2 , f3 ) = c
and
f1 (g1 , g2 , g3 ) = a
f2 (g1 , g2 , g3 ) = b
f3 (g1 , g2 , g3 ) = c
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T. G. Jaiyéo.lá and E. Ilojide
(iii)
An. U.V.T.
HP (Zn ), ∗ is an abelian group if and only if for any Pf , Pg ∈ HP (Zn ),
g1 (f1 , f2 , f3 ) = f1 (g1 , g2 , g3 )
g2 (f1 , f2 , f3 ) = f2 (g1 , g2 , g3 )
g3 (f1 , f2 , f3 ) = f3 (g1 , g2 , g3 )
Proof
The proof is routine.
Theorem 2.3. Let P1 (x, y) = P (x, y) = a + bx + cy represent
a quasigroup
over Zn and let PP (Zn ) = {P1 , P2 , P3 , P4 , P5 , P6 }. Then, PP (Zn ), ∗ is a
subgroup of HP (Zn ), ∗ .
∗
P1
P2
P3
P4
P5
P6
P1
P1
P2
P3
P4
P5
P6
P2
P2
P1
P4
P3
P6
P5
P3
P3
P6
P1
P5
P4
P2
P4
P4
P5
P2
P6
P3
P1
P5
P5
P4
P6
P2
P1
P3
P6
P6
P3
P5
P1
P2
P4
Table 1: Multiplication Table of PP (Zn )
Proof
(Closure) Consider
P2 ∗ P1 = (P2 )1 = (a + cx + by)1 = a + cx + by = P2 ,
P2 ∗ P2 = (P2 )2 = (a + cx + by)2 = a + bx + cy = P1 ,
P2 ∗ P3 = (P2 )3 = (a + cx + by)3 = −ab−1 − b−1 cx + b−1 y = P6 ,
P2 ∗ P4 = (P2 )4 = (a + cx + by)4 = −ab−1 + b−1 x − cb−1 y = P5 ,
P2 ∗ P5 = (P2 )5 = (a + cx + by)5 = −ac−1 + c−1 x − bc−1 y = P4 ,
P2 ∗ P6 = (P2 )6 = (a + cx + by)6 = −ac−1 − bc−1 x + c−1 y = P3 .
Continuing this way, we see that PP (Zn ), ∗ is closed as shown in
Table 1.
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(Associativity) Trivial.
(Identity element) By Table 1, P1 is the identity element.
(Inverse elements) From Table 1 each Pi , i = 1, 2, 3, 4, 5, 6 is invertible.
We conclude that PP (Zn ), ∗ is a subgroup of HP (Zn ), ∗ .
Theorem 2.4. Let P1 (x, y) = P (x, y) = a + bx + cy represent a nquasigroup over Zn and let PP (Zn ) = {P1 , P2 , P3 , P4 , P5 , P6 }. Let S3 = α1 =
o
(1), α2 = (12), α3 = (23), α4 = (123), α5 = (13), α6 = (132) . Define
φ : PP (Zn ), ∗ −→ (S3 , ◦) such that φ(P1 ) = α1 , φ(P2 ) = α2 , φ(P3 ) =
φ
α3 , φ(P4 ) = α4 , φ(P5 ) = α5 and φ(P6 ) = α6 . Then, PP (Zn ), ∗ ∼
= (S3 , ◦)
Proof
The proof of this follows from the definition of φ, Table 1 and Table 2.
◦
α1
α2
α3
α4
α5
α6
α1
α1
α2
α3
α4
α5
α6
α2
α2
α1
α4
α3
α6
α5
α3
α3
α6
α1
α5
α4
α2
α4
α4
α5
α2
α6
α3
α1
α5
α5
α4
α6
α2
α1
α3
α6
α6
α3
α5
α1
α2
α4
Table 2: Multiplication Table of S3
Corollary 2.1. Let P1 (x, y) = P (x, y) = a + bx + cy represent a quasigroup
over Zn and let PP (Zn ) = {P1 , P2 , P3 , P4 , P5 , P6 }. Then,
(i) H1 = {P1 , P2 }, H2 = {P1 , P3 }, H3 = {P1 , P5 }, H4 = {P1 , P4 , P6 } ≤
PP (Zn ).
(ii) HD1 = P2 ,E H2 = P3 , H3 = P5 , H4 = P4 = P6 and PP (Zn ) =
{P4 , P5 } .
(iii)
4
\
Hi = {P1 } and Hi ∩ Hj = {P1 } ∀ i, j = 1, 2, 3, 4, i 6= j.
i=1
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T. G. Jaiyéo.lá and E. Ilojide
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(iv) H1 , H2 , H3 6 PP (Zn ), H4 C PP (Zn ).
(v) H1 H4 = H2 H4 = H3 H4 = PP (Zn ).
(vi) {P1 } = H0 C H4 C PP (Zn ) is a normal series of PP (Zn ).
D
E D
E D
E (vii) PP (Zn )/H4 = P2 H4 = P3 H4 = P5 H4 = P1 H4 , P2 H4 . Hence,
PP (Zn )/H4 is a cyclic group.
(viii) H0 C H4 C PP (Zn ) is a composition series of PP (Zn ).
(ix) PP (Zn ) is a solvable group.
Proof
These can be deduced from Theorem 2.4 and Table 1.
Theorem 2.5. Let P1 (x, y) = P (x, y) = a + bx + cy represent a quasigroup
over Zn and let PP (Zn ) = {P1 , P2 , P3 , P4 , P5 , P6 }. Let fPi =
1
2
3
4
5
6
1(i) 2(i) 3(i) 4(i) 5(i) 6(i)
and
(
SPP (Zn ) =
fPi
)
: PP (Zn ) → PP (Zn ) fPi (Pj ) = Pi ∗ Pj = (Pi )j = Pj(i) .
Define ψ :
PP (Zn ), ∗ −→ SPP (Zn ) , ◦ such that ψ(Pi ) = fPi . Then,
ψ
SPP (Zn ) ≤ S6 and PP (Zn ), ∗ ∼
= SPP (Zn ) , ◦ .
Proof
The proof of these follows from the definition of fPi and ψ.
Corollary 2.2. Let P1 (x, y) = P (x, y) = a + bx + cy represent a quasigroup
over Zn and let PP (Zn ) = {P1 , P2 , P3 , P4 , P5 , P6 }. Define ◦ss on PP (Zn ) ×
PP (Zn ) such that
(Pi , Pj ) ◦ss (Pk , Pl ) = Pi ∗ Pk , Pk ∗ Pj ∗ Pk−1 ∗ Pl
for all (Pi , Pj ), (Pk , Pl ) ∈ PP (Zn ) × PP (Zn ). Then, PP (Zn ) × PP (Zn ), ◦ss
is a Bol loop.
Proof
In Theorem 2.3, PP (Z
n ) is a non-abelian group. Following Theorem 1.6,
PP (Zn ) × PP (Zn ), ◦ss
is a Bol loop.
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References
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[12] G. R. Vadiraja Bhatta and B. R. Shankar, Permutation Polynomials modulo
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[13] W. B. Vasantha Kandasamy, Smarandache loops, Department of Mathematics,
Indian Institute of Technology, Madras, India, 2002
T. G. Jaiyéo.lá
Department of Mathematics,
Obafemi Awolowo University,
Ile Ife 220005, Nigeria.
E-mail: [email protected], [email protected]
E. Ilojide
Department of Mathematics,
Federal University of Agriculture,
Abeokuta 110101, Nigeria.
E-mail: [email protected], [email protected]
Received: 4.04.2012
Accepted: 17.07.2012
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