The Generalization of GCD Matrices

Let 1 2 , , , n S S S L be n finite sets of positive integers and 1 2 n S S S S = × × × L . We have got a q q × matrix S = (gcd( , )) i j q q d d × = , where gcd( , ) ij i j s d d = . In this paper, we study the bounds of the determinant of S , and the value of the determinant of them in special condition. Finally, we generalize the GCD matrices to the direct product of general posets and obtain some results.

. We have got a q q × matrix S = (gcd( , )) , where gcd( , ) In this paper, we study the bounds of the determinant of S , and the value of the determinant of them in special condition.
Finally, we generalize the GCD matrices to the direct product of general posets and obtain some results.

Introduction
The greatest common divisor matrices are a kind of special matrices defined in the positive integer set.And the properties of their determinants have been always the hot research.Beslin and Ligh have defined the greatest common divisor matrix [1] on the positive integer set , , , n S x x x = L , take as ( ) , it is shorten to the GCD matrix and to proved S that is positive definite.In [2] has proved: if S is the FC set then whereφ is the Euler function.In [3], it has defined as the matrix ( ) Smith has proved that if S is FC set, then [4] . Where f μ * is the convolution.In [5-6], some good results have been obtained.In this paper, we have gotten some new generalization in this paper for the GCD matrix.In order to the convenient for the introduction, it is defined as follows.
In [5], It is the definition about the meet semi-lattice, the meet matrix LC and MC set, and LC and MC are the corresponding generalization of FC and GCDC set.
Definition 1 let S be a subset of meet semi-lattice ( , ) U p , defined as , , , ( ) ( ) ( ) , its value is 0 when the sum item is empty , where f is the real function in U , , S f Ψ be called the generalized Euler function on S .

S S S S = × × ×
L , defined a partially ordered relation S p is a finite poset and the meet semi-lattice too., x y S ∀ ∈ , wo have the two conclusions: .Then the meet matrix on S is (gcd( , )) Ψ be the generalized Euler function [5] on i S .It can be defined the generalized Euler function on S as , , , ( , , , ) We have gotten the GCD matrix on S as follows： (gcd( , )) 1 Main results is the minimum GCDC [6] set containing i S .Let T is MC set and , , , , it is clearly to T S ⊇ .We have some conclusions as follows: Theorem1 Let S and T be given above, then ( ( ), ( ), , ( )) and T E are respectively the transpose, T Ψ is the generalized Euler function onT .
And 1 0 Theorem 2 Let S ,T and E be given above, then det S = (det ( , , , )) ( ), ( ), , ( ) is the matrix which has formed by 1 k th， 2 k th，L ， q k th columns taken out E according to the original position.
We can discuss the upper or lower bound of the determinants for S in the following theorem: Theorem 3 Let S , T and E be given above, then(1) det ( ) , the equality holds if and only if S is MC set and the element's divisor in S has no existed in \ T S ; (

Proofs of the main results
In this section, we have mainly proved above theorems.If the positive integer , , , n S x x x = L is the GCDC set, then the generalized Euler function We have given the following lemma.
Lemma 1 Let S andT be given above, then The proof of Theorem 1: Take 1 2 ( ( ), ( ), , ( )) So we have obtained ( ) , ( ) , , ( ) , that is The proof of theorem 2: from the proof of Theorem 1，it is easy to known that Use the Cauchy-Binet formula [9] to T S E E < >= Λ , the result is right.The reason that the third equality has established is that each column has extracted the common divisor in the determinant.So the theorem 2 holds.
function who take the value in the commutative ring R .
we have known that 1 Λ is positive definite.Λ is positive definite, there must be 2 0 E = , that is if and only if S is MC set and the element's divisor in S has no existed in \ T S .T is the GCDC set, so is S .(1) is correct.(2)S is positive definite by theorem 2，we have known that