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Institute of Mathematics and Computer Science, Academy of Sciences of Moldova
Academiei str. 5, Chishinau MD-2028, Republic of Moldova

E-mail: [email protected]

**Received Date:** March 11, 2010; **Revised Date:** April 26, 2010

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We establish connection between product of two matrices of order k k over a eld and the product of the k-mappings corresponding to the k-operations, dened by these matrices. It is proved that, in contrast to the binary case, for arity k 3 the components of the k-permutation inverse to a k-permutation, all components of which are polynomial k-quasigroups, are not necessarily k-quasigroups although are invertible at least in two places. Some transformations with the help of permutations of orthogonal systems of polynomial k-operations over a eld are considered.

It is known that polynomial k-ary operations (shortly, polynomial k-operations), that is, the
operations of the form over a field and systems
of t ≥ k of such operations are used in different applications, in particular in coding theory
and cryptography. If t = k, then we have a matrix of order k × k. However, any
k-tuple (A_{1},A_{2}, . . . . ,A_{k}) of k-operations given on a set Q denes some mapping of the set
Q^{k} into Q^{k} (shortly, a k-mapping):

,

and conversely, any mapping of a set Q^{k} into Q^{k} denes some k-tuple of k-operations [1].

We establish connection between product of two matrices of order k × k over a field and
the product of the k-mappings corresponding to the k-operations, defined by these matrices.
As a corollary, we obtain that the inverse matrix A^{-1} to a nonsingular matrix A is defined
by the components of the k-permutation (that is the bijective k-mapping) inverse to
the k-permutation with the components which are polynomial k-operations, defined by the
matrix A.

In [2], Belousov proved that if A and B are binary quasigroups given on a set Q such
that (A,B) is a permutation of Q^{2}, then the operations C and D, where (C,D) = (A,B)^{-1},
are quasigroups as well. We prove that, in contrast to the binary case, for arity k ≥ 3 the
components of the k-permutation inverse to a k-permutation, all components of which are
polynomial k-quasigroups, are not necessarily k-quasigroups although are invertible at least
in two places.

In different applications, using orthogonal systems of operations, quasigroups, Latin squares, or hypercubes, especially by coding and ciphering of information, necessity to obtain distinct orthogonal systems of operations from one orthogonal system is arisen. In the theory of binary and k-ary operations, some transformations of orthogonal systems of operations (which lead to orthogonal systems) with the help of permutations are known. These trans- formations we use for the most known and often used orthogonal systems of polynomial k-operations (in particular, of polynomial k-quasigroups) over a field.

Recall some necessary designations, definitions, and results.

Let Q be a nite or an innite set, let k ≥ 2 be a positive integer, and let Q^{k} denote the
kth Cartesian power of Q.

A k-groupoid (Q;A) is a set Q with one k-ary operation A defined on Q.

A k-operation B given on a set Q is called isotopic to a k-operation A if there ex-
ists a (k + 1)-tuple of permutations of Q such that, where, shortly B = A^{T} .

A k-ary quasigroup (or a k-quasigroup) is a k-groupoid (Q;A), such that in the equality , each set of k elements from uniquely denes the (k + 1)th element. Sometimes a quasigroup k-operation A is itself considered as a k-quasigroup.

An i-invertible k-operation A defined on Q is a k-operation for which the equation:

has a unique solution for each xed k-tuple .

So a k-ary quasigroup (or simply, a k-quasigroup) is a k-groupoid (Q;A), such that the k-operation A is i-invertible for each i = 1, 2, . . . , k .

The k-operation , on Q with is called the ith identity operation (or the ith selector) of arity k.

Recall also the following information of [1] (for the case k = 2 see [2]).

Let (A_{1},A_{2}, . . . ,A_{k}) (briefly, ) be a k-tuple of k-operations defined on a set Q. This
k-tuple denes the unique mapping in the following way:

,

. These mappings we will call k- mappings.

Conversely, any mapping Q^{k} into Q^{k} uniquely denes a k-tuple of k-operations on, then we define for all. Thus, we obtain the following:

.

If C is a k-operation on Q and is a mapping of Q^{k} into Q^{k}, then the operation defined by the equality is also a k-operation. Let and, then or briefly, . If and are mappings of
Q^{k} into Q^{k}, then according to [1]:

.

If is a permutation of Q^{k}, then .

**Definition 2.1** [1]. A k-tuple of (different) k-operations given on a set Q is called orthogonal if the system has a unique solution for all.

The k-tuple of the selectors of arity k is the identity permutation
of Q^{k} and is orthogonal.

There is a close connection between orthogonal k-tuples of k-operations given on a set
Q and permutations of Q^{k} (such permutations we will call k-permutations) by virtue of the
following

**Proposition 2.2** [1]. A k-tuple of k-operations defined on a set Q is orthogonal if and
only if the mapping is a permutation of Q^{k}.

**Definition 2.3 **[1]. A system , t ≥ k, of k-operations is called
orthogonal if any k-tuple of k-operations of Σ is orthogonal.

**Definition 2.4** [1]. A system , t ≥ 1, of k-operations, given on a set
Q, is called strongly orthogonal if the system is orthogonal.

In a strongly orthogonal system , all k-operations, of Σ are k-
quasigroups since a k-operation A is i-invertible if and only if the mapping (E_{1},E_{2}, . . . ,Ei−1,
A,E_{i+1}, . . . ,E_{k}) is a k-permutation. So the system is called an orthogonal system of k-quasigroups (a k-OSQ) [1].

A k-operation A is a k-quasigroup if and only if the set Σ = {A} is strongly orthogonal. A set of k-quasigroups when k > 2, t ≥ k, can be orthogonal but not strongly orthogonal in contrast to the binary case (k = 2) [1].

Note that in the case of a strongly orthogonal set Σ = {A_{1},A_{2}, . . . ,A_{t}}of k-operations,
the number t of k-operations in Σ can be less than arity k.

Consider k-operations of a special kind (polynomial k-operations), that is k-operations of the form over a field.

A polynomial k-operation is a polynomial k-quasigroup if and only if a_{i} ≠ 0 for all .

If a k-operation B is isotopic to a polynomial k-operation , that is B = A^{T} , where , then

.

Note that the selectors E_{i} of arity k can be also considered as polynomial k-operations over
a field:

, where a_{i} = 1, a_{j} = 0, j ≠ i.

Let a set Σ = {A_{1},A_{2}, . . . ,A_{t}}, k ≥ 2, t ≥ k, be a set of k-operations each of which is a
polynomial k-operation over a field, that is

(3.1)

These polynomial operations dene corresponding rows of the (t × k)-matrix A. It is easy to see from Definition 2.1 that the following statement is valid, where a k-minor is the determinant of a (k × k)-subarray of a matrix A.

**Proposition 3.1 **[3]**.** A system , k ≥ 2, t ≥ k, of polynomial k-operations of (3.1)
is orthogonal if and only if all k-minors of the matrix A defined by these k-operations are
different from 0.

Let in (3.1) t = k, be the (k × k)-matrix the rows of which are
defined by the k-operations A_{1},A_{2}, . . . ,A_{k} and . It is clear that the
mapping is a k-permutation if and only if the matrix A is nonsingular.

The following statement establishes a connection between product of matrices and product of k-mappings.

**Theorem 3.2.** Let A and B be (k×k)-matrices over a field, , let (B_{1},B_{2}, . . . ,B_{k}) be the k-mappings defined by the polynomial k-operations corresponding to
the rows of these matrices, . Then the k-operations C_{1},C_{2}, . . . ,C_{k}
are polynomial and dene the matrix C = AB, that is .

**Proof.** Let then by
the definition,

where .

It means that is the polynomial k-operation defined by the ith row of the matrix AB, so C = AB.

Theorem 3.2 can be formulated otherwise.

**Corollary 3.3.** Let A_{1},A_{2}, . . . ,A_{k} and B_{1},B_{2}, . . . ,B_{k} be polynomial k-operations over a field, , then the k-operations are polynomial,
and the matrix AB is the coecient matrix for them.

**Corollary 3.4.** Let , where A is a nonsingular matrix, and the matrix
A^{−1} is inverse to A. Then is a k-permutation and.

**Proof.** Let . Show thatis the identity permutation of Qk,
that is . Let

Then as it follows from the proof of Theorem 3.2 in the k-operation A_{i}(B_{1},B_{2}, . . . ,B_{k}) the
multipliers by x_{j} , j ≠ i, are equal to 0, and the multiplier by
xi is equal 1, since the operations B_{1},B_{2}, . . . ,B_{k} are defined by the matrix A^{−1}. It means
that but k-selectors E_{1},E_{2}, . . . ,E_{k} dene the rows of the
identity matrix of order k × k the rows of which correspond to the selectors E_{1},E_{2}, . . . ,E_{k}.
Hence, so .

Corollary 3.4 at once implies.

**Corollary 3.5.** If A is a nonsingular (k × k)-matrix and A_{1},A_{2}, . . . ,A_{k} are the polyno-
mial operations defined by the rows of A, respectively, then is a k- permutation, and the permutation denes the polynomial operations
corresponding to the rows of the matrix A^{−1} inverse to A.

It is known that if A and B are orthogonal binary quasigroups given on a set Q, that is
(A,B) is a permutation of Q^{2}, then the operations C and D, where (C,D) = (A,B)^{−1}, are
quasigroups as well (see [2, Lemma 3]). Taking into account this fact and Corollary 3.5, we
obtain that if in a nonsingular matrix A of order 2×2 there is no the element 0, then in the
inverse matrix A^{−1} the element 0 absents also. Below we will show that in the case of arity
k ≥ 3 this statement in general is not true.

The following statement is valid for a k-permutation (A_{1},A_{2}, . . . ,A_{k}) all components A_{i}, of which are polynomial k-ary quasigroups over a field.

**Theorem 3.6.** If all polynomial k-operations A_{1},A_{2}, . . . ,A_{k} over a field are k-quasigroups, is a k-permutation and then each k-operation
of B_{1},B_{2}, . . . ,B_{k} is invertible at least in two places.

**Proof.** A polynomial k-operation is a k-quasigroup if and only if all coecients are distinct from 0. Since
(B_{1},B_{2}, . . . ,B_{k})(A_{1},A_{2}, . . . ,A_{k}) = (E_{1},E_{2}, . . . ,E_{k}), then for any and for all j ≠ i (see the proof of
Corollary 3.4). By Corollary 3.5, all elements of the jth row of the matrix A^{−1} = B cannot
simultaneously be equal to 0 as the matrix A^{−1} = B is nonsingular (by the conditions of the
theorem is a k-permutation).

If k−1 of the coecients b_{j1}, b_{j2}, . . . , b_{jk} is equal 0, then the last coecient is also equal 0.
Thus, there exist at least two elements which are not equal 0 in every row j 6= i of the matrix
B and the k-operation B_{j} , corresponding to it is invertible at least in two places. Changing i,we obtain that the statement is true for any k-operations of B_{1},B_{2}, . . . ,B_{k}.

Using the matrices corresponding to the k-permutations of Theorem 3.6 we obtain the following corollary.

Corollary 3.7. If a nonsingular (k × k)-matrix A has not zero elements, then every row
(every column) of the matrix A^{−1} contains at least two nonzero elements.

**Proof.** This statement for rows follows from Corollary 3.5 and Theorem 3.6. The statement
with respect to columns we can obtain from the proof of Theorem 3.6 considering the prod-
uct (A_{1},A_{2}, . . . ,A_{k}) (B_{1},B_{2}, . . . ,B_{k}) = (E_{1},E_{2}, . . . ,E_{k}), the elementsand reasoning similarly.

Below we will show that the result of Belousov for the binary case, in general, is not true with respect to arity k > 2 (i.e., the result of Theorem 3.6 for k > 2, in general, is not improved) constructing the following two counterexamples of k-permutations for ternary case.

Consider three ternary polynomial operations over the field GF(7):

and three operations:

The following nonsingular (3 × 3)-matrices A, A^{−1} (B,B^{−1}) correspond to these ternary
operations:

The inverse matrix C = A^{−1} (D = B^{−1}) denes the following 3 operations: C_{1}(x, y, z) = 5x+0y+3z, C_{2}(x, y, z) = 4x+6y+4z, C_{3}(x, y, z) = 6x+y+0z (D_{1}(x, y, z) = 6x+6y+3z,
D_{2}(x, y, z) = 0x + 6y + z, D_{3}(x, y, z) = 4x + 6y + 4z). The k-operations C1, C3, D2 are
not 3-quasigroups, but every from them is invertible in two places. The permutation not all components of which are 3-quasigroups, is inverse
to the 3-permutations with quasigroup components.

Now we recall some necessary information from [1] with respect to transformations of or- thogonal systems of k-operations (k-OSOs) (for the case k = 2 see [2]).

Two k-OSOs Σ and given on a set Q are called conjugate if there exists a permutation of Q^{k} such that and a k-OSO is called parastrophic to Σ if where

for any In this case,

By [1, Theorem 1], every k-OSO is conjugate to a k-OSQ, and by [1, Lemma 3], two k-OSQs are conjugate if and only if they are parastrophic.Two orthogonal systems of k-operations Σ and given on a set Q are called isotopic, if is a tuple of permutations of the set Q.

The transformation is called isostrophy.

**Remark 4.1.** Note that if a k-OSO of k-operations on a set Q is strongly
orthogonal (i.e. the system is orthogonal), and is a
(t + k)-tuple of permutations of Q, then , whereare k-quasigroups.

According to [1], the equality is true, that is if then,

(4.1)

In addition, we consider the following case of the transformation of isostrophy of a k-OSO, namely, , where are per-mutations of Q, that is

In this case if then from (4.1), we have

(4.2)

Let and from (4.2), it follows

(4.3)

where

Now consider all these transformations for the case of orthogonal systems of polynomial k-ary operations.

Let be a (t × k)-matrix, let be the orthogonal system of the polynomial k-operations, defined by the corresponding rows of the matrix A (see (3.1)).

**Proposition 4.2.** Let , where are permutations,
then,

Indeed, by the definition of isotopic systems, we have

In this case, the values of the operation A_{i} are changed according to the permutation α_{i},

**Proposition 4.3.** Let wherethen,

If the operations are polynomial and dene a matrix C, then the operations B_{i}, are polynomial and are defined by the matrix AC.

Indeed, in this case,

It is evident that if the operations , are polynomial, then the operations B_{i}, , are also polynomial. Moreover, in this case, the operation Bi is defined by the
ith row of the (t × k)-matrix B = AC (see the form of the operation C_{i} in the proof of
Theorem 3.2 if the matrices B and C change places and ).

**Corollary 4.4.** If in Proposition (D_{1},D_{2}, . . . ,D_{k}), then is an orthogonal system of polynomial k-operations, B_{il} = E_{l}, and

are polynomial k-quasigroups.

**Proof.** By the definition of the transformation of parastrophy, we have since and when but by Corollary 3.4,
all components of the permutation are polynomial k-operations, so all k-
operations of are also polynomial k-operations. Moreover, in this case, we obtain that
the system is strongly orthogonal, and so all k-operations of are polynomial k-quasigroups.

**Proposition 4.5.** If , then,

Indeed, according to (4.3),

**Proposition 4.6.** If , where then

moreover, the operations are k-quasigroups.

This statement follows from Corollary 4.4, Proposition 4.5, and Remark 4.1 since

- Bektenov AS, Jakubov T (1974) Systems of orthogonal n-ary operations (Russian). BulAkadStiince RSS Moldova, SerPhys Math Scipp: 7-14.
- Belousov VD (1968) Systems of orthogonal operations (Russian). Mat Sb 77: 38-58.
- Belyavskaya G, Mullen GL (2005) Orthogonal hypercubesandn-aryoperations.QuasigroupsRelated Systems 13: 73-86.
- Denes J,Keedwell AD (1974) Latin Squares and Their Applications. Academic Press, New York.

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