Constructive Consistent Approximations in Pairwise Comparisons Constructive Consistent Approximations in Pairwise Comparisons

In this paper we investigate groups which admit the existence of weighted consistent approximations for pairwise comparisons matrices. These approximations are deﬁned by extending the classical matrix projection for R + to abstract weighted projections on the non-linear sets of transitive group-valued matrices. It is of interest that all of them are represented by general explicit formulae dependent on an abstract logarithmic function. This general approach is applied to the groups Z p and F 2m m which are of fundamental importance in cryptography. Finally, we use our uniﬁed mathematical model of pairwise comparisons for continuous one-parameter unitary groups, which play a fundamental role in physics. Abstract In this paper we investigate groups which admit the existence of weighted consistent approximations for pairwise comparisons matrices. These approximations are deﬁned by extending the classical matrix projection for R + to abstract weighted projections on the non-linear sets of transitive group-valued matrices. It is of interest that all of them are represented by general explicit formulae dependent on an abstract logarithmic function. This general approach is applied to the groups Z ∗ p and F ∗ 2 m which are of fundamental importance in cryptography. Finally, we use our uniﬁed mathematical model of pairwise comparisons for continuous one-parameter unitary groups, which play a fundamental role in physics. Abstract In this paper we investigate groups which admit the existence of weighted consistent approximations for pairwise comparisons matrices. These approximations are deﬁned by extending the classical matrix projection for R + to abstract weighted projections on the non-linear sets of transitive group-valued matrices. It is of interest that all of them are represented by general explicit formulae dependent on an abstract logarithmic function. This general approach is applied to the groups Z ∗ p and F ∗ 2 m which are of fundamental importance in cryptography. Finally, we use our uniﬁed mathematical model of pairwise comparisons for continuous one-parameter unitary groups, which play a fundamental role in physics.


INTRODUCTION
The consistent approximation plays fundamental role in the pairwise comparisons theory and its applications, cf. Saaty [1,2], Laarhoven and Pedrycz [3], Koczkodaj and Or lowski [4], Cavallo and Brunelli [5], Farkas et al. [6], Koczkodaj and Szarek [7], Holsztyński and Koczkodaj [8], Koczkodaj and Szwarc [9], Koczkodaj et al. [10], Smarzewski and Rutka [11] and other references therein. According to [1,3,4] a consistent projection is defined, for the multiplicative group of positive real and triangular fuzzy numbers, as a composition of additively and multiplicatively invariant matrix mappings. On the other hand, in view of important applications of the pairwise comparisons not only in biology, cryptography and physics, but also in psychology [2], it would be interesting to characterize groups, which admit consistent projections. We note that such attempt has been recently done by Wajch [13]. However, several attempts to define a useful consistent approximation in an abstract setting have not been satisfactorily accomplished until now. An assessment of the latter is discussed in a recent work [14].
In this paper we present a class of groups that admit the existence of non-trivial consistent projections. In doing so, in the next section we give conditions that such groups should satisfy and discuss their properties. Among them the most restrictive and important postulate refers to the existence of appropriate logarithmic and exponential functions. As shown in two sections (Consistent Matrices in Pairwise Comparisons and Abstract Consistent Projections) it is hard to overestimate the influence of abstract logarithms on the generic properties of consistent approximations not only of multiplicative but also of additive type. In addition, both sections in question shed light why the theory of pairwise comparisons is not applicable to every group [13].
The class of abstract logarithmic functions includes the discrete logarithms defined for every cyclic multiplicative group. Therefore, it is possible to derive in Abstract Consistent Projections section the explicit formulae for the weighted consistent projections of reciprocal matrices with entries in the multiplicative groups Z * p = {1, 2, . . . , p−1} of remainders modulo a prime number p. In the next section to Abstract Consistent Projections section we establish results of this kind for multiplicative groups F * 2 m = Z 2 [x]/w m (x) of all non-trivial polynomials over Z 2 = {0, 1} modulo a fixed primitive polynomial w m (x) of degree m with coefficients in the field Z 2 . This suggests that modular consistent projections may become a useful tool on the generic properties of consistent approximations not only of multiplicative but also of additive type. In addition, both sections in question shed light why the theory of pairwise comparisons is not applicable to every group [13].
The class of abstract logarithmic functions includes the discrete logarithms defined for every cyclic multiplicative group. Therefore, it is possible to derive in Abstract Consistent Projections section the explicit formulae for the weighted consistent projections of reciprocal matrices with entries in the multiplicative groups Z * p = {1, 2, . . . , p−1} of remainders modulo a prime number p. In the next section to Abstract Consistent Projections section we establish results of this kind for multiplicative groups F * 2 m = Z 2 [x]/w m (x) of all non-trivial polynomials over Z 2 = {0, 1} modulo a fixed primitive polynomial w m (x) of degree m with coefficients in the field Z 2 . This suggests that modular consistent projections may become a useful tool to to design new algorithms and protocols not only for the computer supported pairwise comparisons, but also for data origin authentication of messages M in cryptography (see e.g. Durnoga and Pomyka la [15]). Finally, we notice that the groups Z * p and F * 2 m already play the central role in several fields of computer science such as e.g. cryptography, processing information, discrete Fourier analysis, coding theory, data storage, modular operations or programming languages. An interested reader may also find some related work in Durnoga andŹra lek [16].
Finally, the second last section is an application of the previous sections to develop pairwise comparisons for one parameter unitary groups U : R → B(H) in the algebra of all bounded linear operators from a Hilbert space H into itself. The results presented there demonstrate the power of functional calculus in designing our model of pairwise comparisons, which is based on using the abstract logarithms. In this model, we construct first an auxiliary additive consistent projection of an antisymmetric matrix [log U (α kj )] n×n onto the group A n (L) of all additively consistent matrices with entries in L = log U (R). In sequel, we use the abstract exponential function to derive a multiplicative consistent projection of a reciprocal matrix M = [m kj ] n×n with entries m kj = U (α kj ) onto the group M n (K) of all multiplicatively consistent matrices T = [t kj ] n×n with entries in K = U (R).

DEFINITIONS AND PRELIMINARIES
A classical concept of pairwise comparisons can be formulated for an algebra A with identity such that A is also an ordered vector space over a field F of real numbers. It means that an order relation ≤ is defined on A, which is reflexive, transitive and invariant with respect to the operations of addition and multiplication by non-negative scalars in F, but not necessarily antisymmetric [17]. In this paper we assume that the order relation is defined on the algebra A by the formula where A + ⊃ {0, 1} is a wedge (a cone) of positive elements in A with vertex 0 and identity 1. For the simplicity such an algebra A will be called ordered.
In the theory of pairwise comparisons there are considered two groups K and L in A: K of multiplicative type and L ⊂ A of additive type. It should be noticed that all pairwise comparisons problems are originally stated for the multiplicative group K. Since these problems are extremely hard to solve even by computers [1], [3], [4], [10] and [11], they are approximated by additive projections onto L-valued consistent matrices, which are eventually transformed by using abstract exponential functions. Definition 1. Two multiplicative and additive groups K and L in an ordered algebra A are said to be logarithmically homeomorphic, if there exists a collection: of homeomorphisms ϕ β : K → L, which have the following properties: and for all x, y ∈ K and α, β ∈ G(K). The homeomorphisms ϕ β will be called abstract logarithms.
Remark 1. If ϕ is an abstract logarithm in Φ(K, L) then it follows from Definition 1 that the inverse mapping ϕ −1 of L onto K has the property: for all u, v ∈ L. Hence the addition in L satisfies the formula: Example 1. The concept of classical pairwise comparisons (see [1], [5], [11] and [18]) in R is characterized by the following pre-assumptions: Note that now condition (2) coincides with the well-known formula log β (α u ) = u log β α.
Example 2. Consider the finite field Z p = {0, 1, . . . , p−1} in which operations of addition and multiplication modulo a prime number p > 2 are introduced. It is clear that Z p becomes an algebra if we set F = Z p . In this case we propose to choose: consists of all generators of the multiplicative group Z * p of positive remainders modulo p and ϕ β (x) = log β x denotes the discrete logarithm of x ∈ Z * p to the base β which is the unique integer y, 0 ≤ y ≤ p − 2, such that β y ≡ x mod p (see [19]). In particular, if we take β = 10 in the set G(Z * 19 ) = {2, 3, 10, 13, 14, 15} of all generators of the multiplicative group Z * 19 , then the elements of additive group Z 18 of discrete logarithms y = log 10 x are as listed in the second row of Table 1. A simple inspection shows that the discrete logarithm y = log 10 x is an homeomorphism between the multiplicative group Z * 19 of remainders modulo 19 and the additive group Z 18 of remainders modulo 18. In particular for x = 7 and y = 16 we have: log 10 (xy) = log 10 112 ≡ log 10 17 (mod 19) = 8 and also log 10 x + log 10 y = 12 + 14 ≡ 8 (mod 18). Remark 2. The finite fields Z p and multiplicative groups Z * p are fundamental in applied cryptography. Additionally, in modern applied cryptography a vital role is also played by cyclic groups of rational points on elliptic curves, c.f. Menezes et al. [19], Husemöller [20] and Washington [21]. Consequently, it would be interesting to investigate the concept of consistent approximation in cyclic groups. In the preliminary step, we address below the latter for the multiplicative groups F * 2 m . Remark 3. Note that the condition (2) from Def. 1 remains true for discrete logarithms. For example, if α = 14, β = 10 and x = 8, then upon resorting to Table 1

CONSISTENT MATRICES IN PAIRWISE COMPARISONS
As in previous section we suppose that K is a multiplicative and L is an additive group in an ordered algebra A over a field F having identity 1 and wedge A + = {x ∈ A : x ≥ 0} ⊃ {0, 1}. In order to formulate the concept of real consistent approximations (cf. e.g. [4], [10] and [1]) in an abstract setting, let K n×n be the set of all K-valued matrices M = [m ij ] n i.j=1 with entries in a multiplicative group K. Then K n×n is a multiplicative matrix group under the operation of pointwise multiplication of matrices M = [m ij ] and The identity in K n×n is matrix E = [e ij ] with all entries e ij = 1. Similarly, we define the additive group L n×n of matrices over an additive group L. In this case the matrix operation is defined by the formulae These matrices are called abstract P C matrices. If an abstract P C matrix M = [m ij ] satisfies in addition the condition of transitivity: then such M is called as consistent. At this point we note that the concept of multiplicative consistency should not be identified with its dual analogue of the abstract additive consistency. Indeed, the latter is defined for the additive matrix group A n×n = A n×n (L), which consists of all antisymmetric matrices A = [a ij ] with entries in the additive group L. More specifically, a matrix A = [a ij ] ∈ A n×n is called additively consistent if it satisfies the following condition: From now on we denote the subgroups of K n×n containing all multiplicative and additive consistent matrices by the following symbols M n = M n (K) and A n = A n (L).
Hence it is antisymmetric. If in addition M is consistent, then the condition of transitivity yields This completes the proof.

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Advances in Science and Technology Research Journal 2022, 16(4), 243-255 Example 3. The inverses of elements in Z * p can be easily computed by the extended Euclidean algorithm [19]. However, if p is small, then it is preferable to use a corollary to Fermat's theorem, which states that the inverse to β k is equal to β p−1−k , where β is a generator of Z * p and k = 0, 1, . . . , p − 1. For the special case of p = 19, all inverses are listed in Table 2

ABSTRACT CONSISTENT PROJECTIONS
According to [1,4,10,12] the classical pairwise comparisons theory is based on the approximation of inconsistent matrices by consistent projections. This approach can be extended to groups M n = M n (K) and A n = A n (L) of consistent matrices as specified below. For this purpose we note that their elements have the following canonical representations (with products xy −1 ∈ K written as ratios x/y) (a) m ij = t i /t j , where t 1 ∈ K is arbitrary and t j = t 1 /m 1j ∈ K, for every j = 2, 3, . . . , n; (b) a ij = s i − s j , where s 1 ∈ ϕ(K) is arbitrary and s j = s 1 − a 1j ∈ ϕ(K), for every j = 2, 3, . . . , n.
Proof. Since t 1 ∈ K and m 1j ∈ K it follows that t j = t 1 /m 1j ∈ K. Hence an easy induction applied to subsequent rows of the multiplicatively consistent matrix M finishes the proof of statement (a): Finally, Theorem 1 and (1) applied to (a) yields: where a ij = ϕ(m ij ) and s j = ϕ(t j ). Indeed, if we take t 1 = 1 and s 1 = 0 then by Table 1 we obtain: Noticeably, both canonical representations of consistent matrices from M n = M n (K) and A n = A n (L) are useful in theory of the abstract consistent approximation. Similarly as in the classical case, one relies here on consistent projections Q n = Q 2 n of M n×n ⊂ K n×n onto M n which are used in turn to approximate abstract P C matrices M ∈ M n×n by their consistent images Q n (M ). Now we introduce a new general concept of weighted consistent projections. In contrast to the classical case, we will not investigate the influence of positive weights ( i ) n i=1 ∈ L on topological properties of these projections. It should be noticed that the sum 1 + 2 + · · · + n of weights may be equal to zero in the case, whenever L is a field of characteristic greater than 0.
Definition 2. Let ϕ be an homeomorphism between logarithmically homeomorphic groups K and L. Let ( i ) n i=1 be a vector of weights in L = ϕ(K) such that and i / are not equal to zero. Then we define mapping Q n : M n×n (K) → M n (K) by the formula: The mappings Q n and P n are said to be consistent. Q n (X · M ) = ϕ −1 (P n (ϕ(X · M ))) = ϕ −1 (P n (ϕ(X)) + P n (ϕ(M ))) = Q n (X) · Q n (M ), guaranteeing that Q n is multiplicative projection and P n is additive projection. Indeed, the latter follows from: The case of K = Z * p and L = Z p−1 coupled with Definition 2 and Lemma 2 leads to: for every matrix M = [m ij ] ∈ M n×n and i = 1, 2 . . . , n.
Proof. The first formula is a direct consequence of Definition 2 for ϕ(x) = log β x and K = Z * p . Therefore, one can apply Remark 3 and (2) in order to obtain: which in turn completes the proof. Visibly, it follows from the congruence 11 · 9 ≡ 4 (mod 19) = 12 that the reciprocal matrix M is not multiplicatively consistent. Upon combining Theorem 2 with Table 1 and Table 2 one arrives at: In order to compute t i = 10 si one can use Table 1 to obtain t 1 = 10 9 ≡ 10 log 10 18 (mod 19) = 18, and similarly t 2 ≡ 7 and t 3 ≡ 14 mod 19. Finally, with the aid of Table 2 The concept of consistent approximation can be generalized for all finite fields, extending the above setting derived to the fields K = Z * p of remainders modulo a prime number p. In doing so, it is sufficient to establish formulae analogous to those presented in Theorem 2. Since each finite field is homeomorphic to a field F p m for some prime p and integer m ≥ 1, we restrict our attention only to the most important in the computer design fields F 2 m of characteristic two, for which the multiplicative groups F * 2 m = F 2 m \ {0} of order N = 2 m − 1 are cyclic. The general case of p > 2 and m > 2, can be treated in a similar, though slightly more complicated manner.
In order to introduce the mentioned above generalization, it is assumed that: is the primitive polynomial of degree m in the ring Z 2 [x] of all polynomials of variable x with coefficients in the field Z 2 , which means that the following two conditions are satisfied: Because of the importance of primitive polynomials w m (x) in applied cryptography, they are listed in the monograph [19] for every m = 1, 2, . . . , 229. Moreover, if 2 m − 1 is a Mersenne prime, then it is done for m less or equal to 44497. We notice that all primitive polynomials w m (x) over Z 2 have the form similar to: The finite field F 2 m = (F 2 m , +, ·) can then be represented as Z 2 [x]/w m (x) defining the set of all polynomials over Z 2 modulo w m (x) with the polynomial β = x forming a generator of F * 2 m . The sum p(x) + q(x) of any two polynomials of degree at most m − 1, has the respective coefficients equal to (a i +b i ) (mod 2), while the product p(x)·g(x) is equal to a remainder of dividing the following polynomial Here a k and b k are taken to be zero if k < 0 or k ≥ m.
Taking into account the latter, one can identify the field F 2 m with the set of all binary numbers of the form:

Then coefficients of the product p(x)q(x) can be determined by a fast algorithm based on the formula
2 is the discrete Fourier transform and the multiplication F (a)·F (b) is understood coordinate-wise [18]. In addition, the inverses and discrete logarithms of elements in F * 2 m = F 2 m \ {0} can be computed by the fast extended Euclidean and square-and-multiply polynomial algorithms [19].

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Advances in Science and Technology Research Journal 2022, 16(4), 243-255 Example 6. The polynomial w 4 (x) = x 4 + x + 1 is primitive over Z 2 . Hence the field F 2 4 = Z 2 [x]/w 4 (x) includes 15 polynomials of the form β k = a 3 x 3 + a 2 x 2 + a 1 x + a 0 , k = 0, 1, . . . , 14, where β = x is a generator in F * 2 4 . Using the decimal notation of F 2 4 , a = (a 3 a 2 a 1 a 0 ) = 8a 3 + 4a 2 + 2a 1 + a 0 ∈ Z 15 , one can list the logarithms and inverses of these polynomials in Table 3. In the next step a theorem on the weighted consistent approximation in the field F q of order q = 2 m is established and is illustrated by a carefully chosen example. For the simplicity the binary representations a = (a m−1 . . . a 0 ) of the members p(x) = m−1 i=0 a i x i of F q are used. Moreover, we recall that matrices in A n and M n have now entries in the additive and multiplicative consistent groups Z q−1 and F * q . Theorem 3. Let β = x = (0 . . . 010) be the generator of the group F * q for q = 2 m , and let ϕ(a) = log β a denote the discrete logarithm of a ∈ F * q to the base β.
for every PC matrix M = [m ij ] ∈ M n×n with the entries in F * q . Proof. Since we have log β (a · b) = (log β a + log β b) (mod (q − 1)) and log β (a x ) = x log β a (mod (q − 1)), x ∈ Z q−1 , we can set ϕ(a) = log β a and K + = F * q in Definition 2 to derive the first formula. The latter directly yields the second formula, as we have: which in turn combined with Lemma 2 completes the proof. 252 Since x · (x 3 + 1) ≡ 1 (mod w 4 (x)) and 1 · 1 = 2 it follows that M is reciprocal, but not consistent. Take now 1 = 3 = 4, 2 = 8 in Z 15 and determine the additive and multiplicative consistent projections: For this purpose, we apply Theorem 2 and Table 3 in order to obtain: s 1 = 4 log β 2 ≡ 4 (mod 15), s 2 = 0, s 3 = 4 log β 9 = 96 ≡ 1 (mod 15), Moreover, with the aid of decimal notation one concludes that polynomials t i = β si , t i /t j ∈ F * 2 4 are equal to: where product 3 · 9 = (0011) · (1001) are computed according to: Finally, consistent projections of the matrix M are equal to:

AN EXTENTION OF CONSISTENT PROJECTION
In Abstract Consistent Projections section the consistent projections Q n : M n×n (K) → M n (K) for all reciprocal matrices with entries in K are defined. It is of considerable interest and particular importance that their range M n×n (K) can be extended to the set of all K -valued matrices, at least when K = F * q with q = 2 m . For this purpose, let β = x = (0 . . . 010) and K n×n (K) denote the generator of F * q = F q \ {0} and the set of all matrices M = [m ij ] with entries in F * q , respectively. Moreover, suppose that ( i ) n i=1 are non-zero weights in Z q−1 such that: Theorem 4. LetQ n : K n×n (K) → M n (K) be a consistent mapping defined by the following formula:

CONCLUSIONS
A study of groups permitting an effective and unified design of weighted projections of abstract reciprocal matrices onto consistent matrices is addressed within the theory of pairwise comparisons. It is shown that this is possible for pairs of multiplicative and additive groups, which are mutually logarithmically homeomorphic. It means that the group of multiplicative type is mapped onto the group of additive type by an abstract logarithm function.
In this paper we have focused our attention mainly on consistent additive and multiplicative projections not only for the discrete modular groups Z * p and F * 2 m but also for one parameter unitary groups in the algebra of all bounded linear operators from a Hilbert space into itself. It has been partially motivated by several applications of these groups in computer sciences, applied mathematics, cryptography and others. By the same reasons it would be of interest to develop pairwise comparisons in other classes of groups, in particular elliptic groups, i.e., groups of rational points on elliptic curves. Finally, we note that another important motivation of study of pairwise comparisons in an abstract setting comes from the field of computer science priority theory.