Generalization of GCD matrices

Special matrices are widely used in information society. The gcd-matrices have be conducted to study over Descartes direct-product of some finite positive integer sets. If Descartes direct-product S=S1×S2×⋯×Sn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ S = S_{1} \times S_{2} \times \cdots \times S_{n} $$\end{document} with n finite positive integer sets as direct product terms, then S is finite too. Without loss of generality, set S=d1,d2,…,dt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ S = \left\{ {d_{1} ,d_{2} ,\ldots, d_{t} } \right\} $$\end{document}, and ∀a=(a1,a2,…,an),b=(b1,b2,…,bn)∈S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \forall {\text{a}} = ({\text{a}}_{1} ,{\text{a}}_{2} ,\ldots, {\text{a}}_{n} ),{\text{b}} = ({\text{b}}_{1} ,{\text{b}}_{2} ,\ldots, {\text{b}}_{n} ) \in S $$\end{document}, the general greatest common factor is defined as gcd(a,b)=∏i=1ngcd(ai,bi)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \gcd ({\text{a}},{\text{b}}) = \prod\nolimits_{i = 1}^{n} {\gcd ({\text{a}}_{i} ,{\text{b}}_{i} )} $$\end{document}. And create a square matrix S=(sij)t×t=(gcd(di,dj))t×t\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \left\langle S \right\rangle = (s_{ij} )_{{{\text{t}} \times {\text{t}}}} = (\gcd (d_{i} ,d_{j} ))_{{{\text{t}} \times {\text{t}}}} $$\end{document} possessed the general greatest common factors gcd(di,dj)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \gcd (d_{i} ,d_{j} ) $$\end{document} as arrays sij=gcd(di,dj)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ s_{ij} = \gcd (d_{i} ,d_{j} ) $$\end{document}. We have researched upper bound and lower bound of the determinant detS\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \det \left\langle S \right\rangle $$\end{document} of the t×t\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ t \times t $$\end{document} gcd-matrix S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \left\langle S \right\rangle $$\end{document}, and compute the determinant’s value under special or specific conditions in the article. At last, some well results about the gcd-matrix has been extend from Descartes direct-product of some finite positive integer sets to general direct product of the posets.


Introduction
The gcd-matrix ⟨S⟩ standing for the greatest common divisor matrix is a special type of matrix that the arrays come from a positive integers set S = a 1 , a 2 , … , a n . The nature of its determinants det ⟨S⟩ has increasingly become a hot topics of scientific research in many application fields. The authors such as S. Beslin, S. Ligh etc. have taken the lead in giving a definition of the greatest common factor matrix [4,18,19,22,23] (that is gcd-matrix ⟨S⟩ ) over the finite natural number set S = a 1 , a 2 , … , a n . Set the square matrix ⟨S⟩ = (s ij ) n×n with the arrays s ij = gcd(a i , a j ) , the array s ij = gcd(a i , a j ) is the greatest common factor between a i and a j . The greatest common factor (or greatest common divisor)is abbreviated to GCD. And that naturally gives rise to the concept of a gcd-matrix ⟨S⟩ . Z. Li has pointed out det ⟨S⟩ = (a 1 ) (a 2 ) ⋯ (a n ) in [14,26], where S is the FC set and is the Euler's totient function. While the set is said a FC set, if the all positive integer factors of a ∈ S are contained in the set S . Namely, if ∀a ∈ S and d|a then d ∈ S . The matrix ⟨S⟩ g has been defined as ⟨S⟩ g = (s ij ) n×n ,s ij = g((a i , a j )) (i, j = 1, 2, … , n) and g((a i , a j )) refers to the value of the arithmetic function g on the greatest common factor gcd(a i , a j ) [1,2,7,8,12,13,21]. The following conclusions are proved by H.J.S. Smith [7]: if the set S is hypothetical to FC, then completely satisfactory formula det ⟨S⟩ g = (g ⊙ )(a 1 )(g ⊙ )(a 2 ) ⋯ (g ⊙ )(a n ) holds, where g ⊙ just happens to be the convolution between g and the mobius function . The conclusions in [7,26] have been extended respectively the corresponding promotion, and they have obtained some good results [5,10,11]. This article has obtained some new promotion to the gcd-matrix based on the key outcomes in [17, 24,25]. These properties are widely used in the communication theory, the algebraic coding theory, the cryptography and other fields [3,9,15]. For the convenience of introduction, the following definitions have been given first.
Definition 1 [25] Let U be a poset, p be its partial order. Called U the meet semilattice, when ∀a, b ∈ U , there exists the unique element ∈ U . The element satisfies: (i) pa the value 0 when the subscript sum item is null set. Set g is the real function denoted by g ∶ U → R , S,g be known as the generalized euler's -function on the partial order subset S.

Remark 1
It is easy to know from definition 4, the backstepping formula g(x j ) = ∑ x i px j S,g (x i ) sets up when S is a partial order subset of the meet semi-lattice U, then the function come from the mobius inversion [20], and is the mobius function on the partial order subset S with the partly ordered relation p.
As previously defined, the concept of the gcd-matrix has been generalized onto the Descartes direct product. There are n positive integer finite sets S 1 , S 2 , … , S n , for all 1 ≤ i ≤ n , there exists the partly ordered relation p i belonged to the partial order subset S i , namely ∀a i , b i ∈ S i , a i p i b i if and only if a i |b i denoted by the divisible relation of natural ⋮ ⋮ ⋱ ⋮ h((a n , a 1 ) p ) h((a n , a 2 ) p ) ⋯ h((a n , a n ) p ) ⎞ ⎟ ⎟ ⎟ ⎠ n×n numbers. So (S i , p i ) be the partial order set. Using Definition 1, the natural number set N can become a meet semilattice under the condition of the partly ordered relation is divisibility relation. Moreover, the meet of any two natural numbers is the greatest common divisor. So (S i , p i ) is viewed as a subset that can be embedded in some meet semilattice. We have the result: And set S = S 1 × S 2 × ⋯ × S n , defined a divisible relation as the partially ordered p on the natural number set S. For all ∀a = (a 1 , a 2 , … , a n ), b = (b 1 , b 2 , … , b n ) ∈ S , make apb ⇔ a i |b i (i = 1, 2, … , n) . Still is denoted by (S, p) a poset. Every set S i is finite set, so is S. Without loss of generality, let S = d 1 , denotes the cardinality of the set S i . At the same time, ∀a, b ∈ S , we have two points as follows: (i) if a|b ⇔ apb , a is called one factor of b.
So (S, p) viewed as a subset can be embedded in some meet semilattice denoted by N × N × ⋯ × N , fulfilled The q × q meet-matrix over S is defined as ⟨S⟩ = (gcd(d i , d j )) q×q . In order to vivid charm, ⟨S⟩ is still known as the gcd-matrix. Investigators have found out that the gcd-matrix and generalized Euler's -function have close relationships.
Let S i be the broad sense Euler's totient function over the We have defined a generalized Euler's S -function over the set S as follows: j note the product of the component in d j . Now, we can study the gcd-matrix on is a n-dimensional vector. We have gotten a gcd-matrix over S = S 1 × S 2 × ⋯ × S n = d 1 , d 2 , … , d q as follows: .
After we have obtained the gcd-matrix on S = S 1 × S 2 × ⋯ × S n = d 1 , d 2 , … , d q , the nature of the gcd-matrix have been researched in next section. In this paper, we have two generalizations: first, we extend the GCD matrix to the Descartes direct-product of some finite positive integer sets and the general direct product of the posets; second, we define the generalized Euler's totient-function. And the relation between the determinant of the generalized GCD matrix and the generalized Euler's totient-function is studied.

Main results
Next, we have studied the generalized gcd-matrix (or called meet matrix) over S = S 1 × S 2 × ⋯ × S n used the Mobius inversion. Set T 1 , T 2 , … , T n finite natural number sets, T i (i = 1, 2, … , n) denote a minimum GCDC-set containing the set S i (where the smallest GCDC set refers to be contained in the intersection of all GCDC subsets of S i ). As previously mentioned, T i is a poset under the divisibility relation. Still the partially ordered relation of T i denotes by We have drawn the following conclusions to the generalized Euler's totient -function: . The matrices A T and E T are respectively the transpose matrices to A and E, T is the generalized Euler's totient function on T. And According to this theorem, we can calculate the determinant of ⟨S⟩ , and have the following theorem: Descartes direct product of n sets, so the determinant is expanded as follows: the sub-matrix E(k 1 , k 2 , … , k q ) has been formed by taking out k 1 th, k 2 th,…, k q th columns of the 0-1 matrix E and retaining the initial corresponding position unchanged. By the following Theorem 3, We can have a discussion on upper-lower bounds of the meet-matrix's determinant, it is expressed the following theorem.
, its equality is true iff the set S is MC, and the factor of any one element in S does not exist in the difference set T\S; It is a special case of the third item in the Theorem 3. We have a certain understanding to the greatest common divisor matrix on S = S 1 × S 2 × ⋯ × S n , especially the structure and its determinants of the matrices.

Proofs of the above theorems
Starting from this part, the authors have priority task to prove the 3 theorems mentioned above, it is to need the following proposition: Proposition 1 [20] Let the set S = a 1 , a 2 , … , a n be partial order, and p be the partially ordered relation, then S may be orderly arranged a 1 , a 2 , … , a n , and if a i pa j then i < j.
Proposition 2 [20] Let the finite set S = a 1 , a 2 , … , a n be GCDC, the set T = y 1 , y 2 , … , y n included the set S be FC, if a 1 < a 2 < ⋯ < a n , then S (a j ) = ∑ z∈V a j (z), ∀j ≥ 1 , where the subscript set V a j = z ∈ T ∶ min{y ∈ S ∶ z|y} = a j and take as a 0 = 0.

Remark 2
The proposition has indicated that if the natural number set S = a 1 , a 2 , … , a n is GCDC, then the broad sense Euler's totient T -function S (a) ≥ (a), ∀a ∈ S , where is the Euler's function. When the set S is FC, so S (a) = (a), ∀a ∈ S , b e c a u s e o f S = T a n d V a j = z ∈ T ∶ min{y ∈ S ∶ z|y} = a j = a j at this time.
Proposition 3 [16] Let the two matrices C and D be n order real number symmetric, if the n × n matrix C is positive definite, and the n × n matrix D is positive semi-definite, then the determinant satisfies det(C + D) ≥ det C + det D , when the equality establishes iff the order n ≥ 1 and D is a zero matrix.
Proposition 4 [20] Let be the Mobius function on the locally finite poset , then (i) When is an isomorphic mapping from the poset to P, (a, b) = P ( (a), (b))(a, b ∈ ). (ii) In addition, we have given two following lemmas. 1 , a 2 , … , a n ) ∈ S , then the general Euler's totient S -function: Proof Let S = d 1 , d 2 , … , d q be Descartes direct product, and the Euler's totient S -function over S is generalized, we know as the product d (1) j . In order to profile, ∀d ∈ S written the product |d| ≜ d (1)  The proof process of Theorem 1 Take = diag( T (f 1 ), Hence we have gained < S >= E E T . From Lemma 1, it comes very naturally to get the formula S (a) = ∏ n i=1 S i (a i ) . From proposition 2, the set T k (k = 1, 2, … , n) is GCDC, when the function satisfies T k ≥ (y k ) > 0 , so the Euler's , then < S >= AA T , that is to say < S >= AA T = E E T . So perfect is to the Theorem 1. The proof process of Theorem 2 is as follows: Proof of Theorem 2 It is easy from the proof process of Theorem 1 to know the following results < S >= E E T . Take the matrix D = E = e ij T (f j ) q×t . By the Cauchy-Binet formula [6] to decompose matrix < S >= E E T .
The third equation is established because each column extracts the common factor in the following determinant det D.
Thereupon then the Theorem 2 is correct.□

Corollary 1 If the set
Proof By Proposition 1, we have known that if d i pd j and d i ≠ d j ⇒ i < j to S. By Theorem 2, the matrix E = e ij is the lower triangular matrix that the main diagonal are all 1. That is if i < j then e ij = 0 and e ii = 1 . The lower triangular matrix is So is correct to the conclusion. □ The proof process to Theorem 3 It is easy from Theorem 2 to know that < S >= E E T and = diag( T (f 1 ), T (f 2 ), … , T (f t )) .
According to the Theorem 2, the inequality holds.
To consider the sufficient and necessary condition of the equality to hold, by proposition 1, Set the poset We have the following conclusion of the determinant det ⟨S⟩ = det(M + N) ≥ detM + detN ≥ detM by Proposition 3. Namely the inequality det ⟨S⟩ ≥ ∏ d∈S T (d) holds, the formula det ⟨S⟩ ≥ ∏ d∈S T (d) has established iff N = 0 here. While 2 is a positive-definite matrix, it must satisfy E 2 = 0 , namely the poset S is "the MCS" and the factor of elements included in the poset S doesn't exist in the difference set T\S. The set T is GCDC, which make the common divisor of any two elements in S to include in S, so is GCDC to S. The conclusion (i) is right.
(ii) According to Theorem 2, the matrix ⟨S⟩ is positive-definite. We have arrived at a conclusion based on Lemma 2.
If the poset S is FC, then the two sets are equal, namely S = T . We have known T k (y k ) = (y k ) by Remark 2, therefore, Where is the Euler's totient -function. To make a long story short, the conclusions (i) (ii) (iii) are all hold, and the proof of Theorem 4 have been finished now. □

Remark 2
It is clearly to the above process for S = S 1 × S 2 × ⋯ × S n that, if n = 1 , then it can get the content discussed in [7]; if n = 2 that is S = S 1 × S 2 , then ⟨S⟩ = ⟨S 1 ⟩ ⊗ ⟨S 2 ⟩,where ⊗ is the tensor product.

The generalization of gcd matrix on a general poset
Many properties of gcd-matrices are related to the ordered relations of set, we can study generalized gcd-matrix on the general partial order set. In order to discuss more properties of the gcd-matrix and meet matrix, we have studied the meet matrix from the direct product of a general poset. Set the n sets P 1 , P 2 , … , P n are all meet semilattices, naturally occurring respective partialy ordered relation. The notation p i (i = 1, 2, … , n) is expressed the corresponding partial order relation of P i , then P = P 1 × P 2 × ⋯ × P n is also a poset and the meet semi-lattice with the partial order relation p. ∀a = (a 1 , a 2 , … , a n ), b = (b 1 , b 2 , … , b n ) ∈ P , apb ⇔ a i p i b i (i = 1, 2, … , n).
T h e n t h e m e e t b e t w e e n x a n d y i s (a, b) p = ((a 1 , b 1 ) p 1 , (a 2 , b 2 ) p 2 , … , (a n , b n ) p n ).
Set g i ∶ P i → R a function with the value come from an Abelian ring R. Let the vector function g ∶ P → R be a function over a meet semilattice P, ∀a = (a 1 , a 2 , … , a n ) ∈ P , defined g(a) = ∏ n i=1 g i (a i ) . It is apt to the product access to the Abelian ring R.
Set S i the finite partial order subset of the poset P i , and T i the smallest MC finite subset of the poset P i including in S i . If the direct product S = S 1 × S 2 × ⋯ × S n and T = T 1 × T 2 × ⋯ × T n , then the set T is still MC, including in the subset S.
Set the Euler's totient function S i ,g i over set S i is a component function of the broad sense Euler's totient S,g -function, then the S,g is the broad sense Euler's totient S i ,g i det ⟨S⟩ = � d∈S S (d) = � (a 1 ,a 2 ,…,a n )∈S [ (a 1 ) (a 2 ) ⋯ (a n )] -function defining over Descartes direct product S. In addition,S i , T i are finite subsets, so S, T are finite subsets too. Without the generality, let S = d 1 , d 2 , … , d q , Similar to the previous Lemma 1, ∀d = (d 1 , First, we have discussed the structure of ⟨S⟩ g , in order to facilitate discussion, it is always assumed that is ⟨S⟩ g positive define, so S,g need to be restricted.
We have restricted the Abelian ring R to the real number field with g i ∶ P i → R , at the same time, let S i ,g i > 0.
We have known when the meet matrix ⟨S⟩ g is positive definite, it is easy to get the properties of the meet-matrix over Descartes direct-product of a general poset.

the Descartes direct-product, the set T is MC and containing the subset S, when Euler's totient function
, the equality is fulfilled iff the poset S is MC. Meanwhile, there no exist the element in the difference set T\S that has fulfilled the partial order relationships p with any one element in the poset S.
(ii) (Determinant's upper bound) Set the strong inequality det ⟨S⟩ g < ∏ d∈S g(d) is establishment. That theorem mentioned earlier is necessary about S,g > 0 , which is easier to discuss to make ⟨S⟩ g positive definite. But the following theorem is not necessary about S,g > 0 , so we can get the following results:  3 have gotten the same conclusion on the determinant of a meet-matrix, so its proof is omitted. Theorem 5 is also same argument with the Theorem 13 [7]. Similar processing, we won't talk more about the proof here.

Conclusions
The gcd-matrices on some sets are a special class of matrices with many beautiful properties. Their determinant calculations are simple and quick, and it helps to design lightweight cryptographic arithmetics and key exchange protocol [27][28][29]. In addition, the determinant calculation of gcdmatrix is easy to program because of its low computational complexity. Because of its wide application, the gcd-matrix is a valuable long-term research topic. The mixcolumn cryptography characteristics and the maximum branchnumber of gcd-matrix are important research parameters.
The generalized greatest common factor matrices (ggcdmatrix) are a class of special importance and perfection matrices, this article has studied the value of their determinants. We have known that the consummate mathematics relations between generalized greatest common divisor matrices and the generalized Euler's totient S,g − function, these relations are taken into account a construction perspective. Our next work goal is to study the eigenvalues, the invariant factor and the characteristic divisor for the generalized greatest common divisor matrix (ggcd-matrix) on the general poset S. The relationship between the branch number of meet matrices and the generalized Euler's totient S,g − function needs further study. Using meet matrices as the generating matrix to design error correcting codes is an important research field of algebraic coding. That kind of error-correcting code can be constructed the quantum cryptography resisting quantum computation attacks, which is also an important research area of quantum communication.