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Polynomial k-ary operations, matrices, and k-mappings | OMICS International
ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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Polynomial k-ary operations, matrices, and k-mappings

G. BELYAVSKAYA

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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Abstract

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, de ned 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.

1 Introduction

It is known that polynomial k-ary operations (shortly, polynomial k-operations), that is, the operations of the form Image 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 Image of order k × k. However, any k-tuple (A1,A2, . . . . ,Ak) of k-operations given on a set Q de nes some mappingImage of the set Qk into Qk (shortly, a k-mapping):

Image,

and conversely, any mapping of a set Qk into Qk de nes 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) Image inverse to the k-permutation Imagewith 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 Q2, 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.

2 Preliminaries

Recall some necessary designations, definitions, and results.

Let Q be a nite or an in nite set, let k ≥ 2 be a positive integer, and let Qk 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 Image of Q such thatImageImage, whereImage, shortly B = AT .

A k-ary quasigroup (or a k-quasigroup) is a k-groupoid (Q;A), such that in the equality Image, each set of k elements fromImage uniquely de nes 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:

Image

has a unique solution for each xed k-tuple Image.

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 Image.

The k-operation Image, on Q with Image 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 (A1,A2, . . . ,Ak) (briefly, Image) be a k-tuple of k-operations defined on a set Q. This k-tuple de nes the unique mapping Image in the following way:

Image,

Image. These mappings we will call k- mappings.

Conversely, any mapping Qk into Qk uniquely de nes a k-tuple Image of k-operations onImage, then we defineImage for allImage. Thus, we obtain the following:

Image.

If C is a k-operation on Q and Image is a mapping of Qk into Qk, then the operation Image defined by the equalityImage is also a k-operation. Let Image andImage, thenImage or briefly, Image. If Image and Image are mappings of Qk into Qk, then according to [1]:

Image.

If Image is a permutation of Qk, then Image.

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

The k-tuple Image of the selectors of arity k is the identity permutation of Qk and is orthogonal.

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

Proposition 2.2 [1]. A k-tuple Imageof k-operations defined on a set Q is orthogonal if and only if the mapping Image is a permutation of Qk.

Definition 2.3 [1]. A system Image, 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 Image, t ≥ 1, of k-operations, given on a set Q, is called strongly orthogonal if the system Image is orthogonal.

In a strongly orthogonal system Image, all k-operationsImage, of Σ are k- quasigroups since a k-operation A is i-invertible if and only if the mapping (E1,E2, . . . ,Ei−1, A,Ei+1, . . . ,Ek) is a k-permutation. So the system Image 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 Image 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 = {A1,A2, . . . ,At}of k-operations, the number t of k-operations in Σ can be less than arity k.

3 Product of (k × k)-matrices and product of k-mappings

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

A polynomial k-operation is a polynomial k-quasigroup if and only if ai ≠ 0 for all Image.

If a k-operation B is isotopic to a polynomial k-operation Image, that is B = AT , where Image, then

Image.

Note that the selectors Ei of arity k can be also considered as polynomial k-operations over a field:

Image, where ai = 1, aj = 0, j ≠ i.

Let a set Σ = {A1,A2, . . . ,At}, k ≥ 2, t ≥ k, be a set of k-operations each of which is a polynomial k-operation over a field, that is

Image (3.1)

These polynomial operations de ne 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 Image, 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, Image be the (k × k)-matrix the rows of which are defined by the k-operations A1,A2, . . . ,Ak and Image. It is clear that the mapping Image 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, Image, let Image (B1,B2, . . . ,Bk) be the k-mappings defined by the polynomial k-operations corresponding to the rows of these matrices, Image. Then the k-operations C1,C2, . . . ,Ck are polynomial and de ne the matrix C = AB, that is Image.

Proof. Let Image then by the definition,

Image

where Image.

It means that Image 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 A1,A2, . . . ,Ak and B1,B2, . . . ,Bk be polynomial k-operations over a field, Image, then the k-operations Image are polynomial, and the matrix AB is the coecient matrix for them.

Corollary 3.4. Let Image, where A is a nonsingular matrix, and the matrix A−1 is inverse to A. Then Image is a k-permutation andImage.

Proof. Let Image. Show thatImageis the identity permutation of Qk, that is Image. Let

Image

Then as it follows from the proof of Theorem 3.2 in the k-operation Ai(B1,B2, . . . ,Bk) the multipliers by xj , j ≠ i, are equal to 0, and the multiplier Image by xi is equal 1, since the operations B1,B2, . . . ,Bk are defined by the matrix A−1. It means that Imagebut k-selectors E1,E2, . . . ,Ek de ne the rows of the identity matrix of order k × k the rows of which correspond to the selectors E1,E2, . . . ,Ek. Hence, Imageso Image.

Corollary 3.4 at once implies.

Corollary 3.5. If A is a nonsingular (k × k)-matrix and A1,A2, . . . ,Ak are the polyno- mial operations defined by the rows of A, respectively, then Image is a k- permutation, and the permutationImage de nes 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 Q2, 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 (A1,A2, . . . ,Ak) all components Ai, Imageof which are polynomial k-ary quasigroups over a field.

Theorem 3.6. If all polynomial k-operations A1,A2, . . . ,Ak over a field are k-quasigroups, Image is a k-permutation and Image then each k-operation of B1,B2, . . . ,Bk is invertible at least in two places.

Proof. A polynomial k-operation Imageis a k-quasigroup if and only if all coecients Image are distinct from 0. Since (B1,B2, . . . ,Bk)(A1,A2, . . . ,Ak) = (E1,E2, . . . ,Ek), then Image for any Imageand Image 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 Image is a k-permutation).

If k−1 of the coecients bj1, bj2, . . . , bjk 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 Bj , corresponding to it is invertible at least in two places. Changing i,Imagewe obtain that the statement is true for any k-operations of B1,B2, . . . ,Bk.

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 (A1,A2, . . . ,Ak) (B1,B2, . . . ,Bk) = (E1,E2, . . . ,Ek), the elementsImageImageand 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):

Image

and three operations:

Image

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

Image

The inverse matrix C = A−1 (D = B−1) de nes the following 3 operations: C1(x, y, z) = 5x+0y+3z, C2(x, y, z) = 4x+6y+4z, C3(x, y, z) = 6x+y+0z (D1(x, y, z) = 6x+6y+3z, D2(x, y, z) = 0x + 6y + z, D3(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 ImageImagenot all components of which are 3-quasigroups, is inverse to the 3-permutations Image with quasigroup components.

4 Transformations of orthogonal systems of polynomial k-operations

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 Imagegiven on a set Q are called conjugate if there exists a permutation Image of Qk such that Imageand a k-OSO Imageis called parastrophic to Σ if Image where

Image for anyImage In this case,

Image

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 Imagegiven on a set Q are called isotopic, if Imageis a tuple of permutations of the set Q.

The transformation Image is called isostrophy.

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

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

Image (4.1)

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

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

Image(4.2)

Let Image and from (4.2), it follows

Image(4.3)

where Image

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

Let Imagebe a (t × k)-matrix, let Image 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 Image , whereImage are permutations, then,

Image

Indeed, by the definition of isotopic systems, we have

Image

In this case, the values of the operation Ai are changed according to the permutation αi, Image

Proposition 4.3. Let Image whereImagethen,

Image

If the operations Image are polynomial and de ne a matrix C, then the operations Bi, Imageare polynomial and are defined by the matrix AC.

Indeed, in this case,

Image

It is evident that if the operations Image, are polynomial, then the operations Bi, Image, 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 Ci in the proof of Theorem 3.2 if the matrices B and C change places and Image).

Corollary 4.4. If in Proposition Image (D1,D2, . . . ,Dk), then Image is an orthogonal system of polynomial k-operations, Bil = El, Imageand

Image

are polynomial k-quasigroups.

Proof. By the definition of the transformation of parastrophy, we have ImageImage since Image and ImageImagewhen Imagebut by Corollary 3.4, all components Imageof the permutation Image are polynomial k-operations, so all k- operations of Imageare also polynomial k-operations. Moreover, in this case, we obtain that the system Imageis strongly orthogonal, and so all k-operations of Imageare polynomial k-quasigroups.

Proposition 4.5. If ImageImage, then,

Image

Indeed, according to (4.3), ImageImage

Proposition 4.6. If Image, where ImageImage then Image

Image

moreover, the operations Image are k-quasigroups.

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

Image

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