最大公因数闭集上幂矩阵的行列式整除性

设S={x1,…,xn)是由n个不同正整数组成的最大公因数闭集,我们证明: (1)如果n≤3,则对(?)ε∈Z+,有det(S)nε整除det[S]nε;(2)如果maxxi∈S{xi}<12, 则对(?)ε∈Z+,有det(S)nε整除det[S]nε;(3)如果maxx∈S{R(x)}≤1,其中R(x)是x 在S中的最大型因子集,则对(?)ε∈Z+,有det(S)nε整除det[S]nε．     

定义在广义Gcd—closed集S上的GGCD矩阵和GLCM矩阵的逆

设S＝｛x1,x2,…xn｝是一个由非零整数且|xi|≠|xj≠k.1≤i,j≤n)组合的集合,我们先定义了集S上的广义GCD（GGCD）矩阵和广义LCM（GLCM）矩阵,然后计算了定义在广义gcd-closed集上的GGCD矩阵和CLCM矩阵的逆矩阵。     

有限幂零半群的构造

有零元的半群S称为幂零的,如果对于任意x∈S有正整数n使x~n=0,设S是有限幂零半群,则存在正整数门使得S~n={0}([1],P.81),适合此条件的最小正整数称为S的0-阶,记为0(S);S的非零元x的0-阶(记为o(x))是指适合下述条件的最小正整数n:S的任意n个包括x在內的元素的积为零;零元的o-阶规定为1. 以下假设S是n阶有限幂零半群.     

不可约与几乎可约布尔矩阵的幂敛指数

§1.引言 布尔矩阵是指元素按如下规则运算的(0,1)矩阵:a+b=max{a,b},a·b=min{a,b}(a,b∈{0,1}),n阶布尔方阵的集合记为B_n。一个布尔方阵A的幂敛指数k(A)是满足如下条件的最小非负整数k: 条件:存在正整数p,使A~k=A~(k+p), (1.1)而称满足条件A~(k(A))=A~(k(A)+p)的最小正整数p为A的周期,记作p(A)。 对布尔矩阵的幂序列及幂敛指数的研究在有限自动机理论、二元关系理论及遍历指     

混合幂为2,3,a和b的丢番图不等式

运用Davenport-Heilbronn方法证明了如果η是实数,λ1,μ1,μ2,μ3,μ4,θ1,θ2是非零实数,并且不同一符号,且至少一个λ1/μi(i=1,2,3,4)是无理教,假设(i)a=3,3≤b≤11,或者(ii)a=4,4≤b≤5,那么对某些σ=σ(a,b)>0,混合幂为2,3,a和b的丢番图不等式有无穷多正整数解x1,...,x7.     

六元gcd封闭集上Smith矩阵的整除性

我们给出了关于六元gcd封闭集S的充分必要条件,使得在整数矩阵环M_6(Z)中,定义在S上的e次幂GCD矩阵(S~e)整除e次幂LCM矩阵[S~e].这部分解决了Hong在2002年提出的一个公开问题.     

圈布尔矩阵类的幂敛指数集

在布尔运算下, 布尔矩阵A的幂敛指数和周期分别是使A^k=A^k+p成立的最小非负整数k和最小正整数p. 人们对周期的认识已经相当完善.给定满足一个不等式的正整数n和s, 利用组合分析确定了有向图含至少一个s -圈的n×n布尔矩阵的幂敛指数可以取得的数值.     

一类幂零矩阵幂零指标的特征

设(L, ,·)是一个incline. 本文给出了一个incline上幂零矩阵幂零指标的特征.其结果改进了文[4]中的相应结论.     

LCM矩阵的可逆性与不定方程ayz+bzx+cxy=xyz+1的解

设S＝{x1,x2,…xn}是不同正整数的集合。已经知道当n≤7时在最大公因数封闭集S上的LCM矩阵是可逆的；也知道当n≥9时有无限多个包含整数1的最大公因数封闭集它们的LCM矩阵是奇异的；这篇文章的主要结果是证明当n=8且包含整数1时，除了20个最大公因数封闭集外，其余所有最大公因数封闭集上的LCM矩阵都是可逆的，而这归结为解一个不定方程。     

一般环之因子幂零理想

本文称环Ω的左(右)理想A为因子幂零的,如果对于任意元素r∈Ω,均有正整数m=m(r),使得A^mr={0}.称Ω的一个左理想L为关于元素b∈Ω的左因子,如果Lb≠{0}.定理4 设R是环Ω的因子幂零右理想,那么R+ΩR是Ω的一个因子幂零理想.定理7 设Ω具有局部左因子极小条件,那么Ω的任意诣零左理想必是因子幂零左理想.本文指出因子幂零性是介于幂零性与诣零性之间的一种性质,更接近幂零性。     

GCD和GCUD矩阵行列式的一个不等式（英）

Let S = {x₁, x₂,..., x_n} be a set of distinct positive integers. The n x n matrix (S) whose i, j-entry is the greatest common divisor (x_i, x_j) of x_i and x_j is called the GCD matrix on S. A divisor d of x is said to be a unitary divisor of x if (d, x/d) = 1. The greatest common unitary divisor (GCUD) matrix (S^**) is defined analogously. We show that if S is both GCD-closed and GCUD-closed, then det(S^**) ≥ det(S), where the equality holds if and Only if (S^**) = (S).     

Divisibility among power GCD matrices and among power LCM matrices on three coprime divisor chains

Let h be a positive integer and S?=?{x ₁,?…?,?x _h } be a set of h distinct positive integers. We say that the set S is a divisor chain if x _σ(1) ∣?…?∣ x _σ(h) for a permutation σ of {1,?…?,?h}. If the set S can be partitioned as S?=?S ₁?∪?S ₂?∪?S ₃, where S ₁, S ₂ and S ₃ are divisor chains and each element of S _i is coprime to each element of S _j for all 1?≤?i?<?j?≤?3, then we say that the set S consists of three coprime divisor chains. The matrix having the ath power (x _i , x _j ) ^a of the greatest common divisor of x _i and x _j as its i, j-entry is called the ath power greatest common divison (GCD) matrix on S, denoted by (S ^?a ). The ath power least common multiple (LCM) matrix [S ^?a ] can be defined similarly. In this article, let a and b be positive integers and let S consist of three coprime divisor chains with 1?∈?S. We show that if a?∣?b, then the ath power GCD matrix (S ^?a ) (resp., the ath power LCM matrix [S ^?a ]) divides the bth power GCD matrix (S ^?b ) (resp., the bth power LCM matrix [S ^?b ]) in the ring M _h (Z) of h?×?h matrices over integers. We also show that the ath power GCD matrix (S ^?a ) divides the bth power LCM matrix [S ^?b ] in the ring M _h (Z) if a?∣?b. However, if a???b, then such factorizations are not true. Our results extend Hong's and Tan's theorems and also provide further evidences to the conjectures of Hong raised in 2008.     

Divisibility of determinants of power GCD matrices and power LCM matrices on finitely many quasi-coprime divisor chains

Let a, n ? 1 be integers and S = {x₁, … , x_n} be a set of n distinct positive integers. The matrix having the ath power (x_i, x_j)^a of the greatest common divisor of x_i and x_j as its i, j-entry is called ath power greatest common divisor (GCD) matrix defined on S, denoted by (S^a). Similarly we can define the ath power LCM matrix [S^a]. We say that the set S consists of finitely many quasi-coprime divisor chains if we can partition S as S = S₁ ∪ ? ∪ S_k, where k ? 1 is an integer and all S_i (1 ? i ? k) are divisor chains such that (max(S_i), max(S_j)) = gcd(S) for 1 ? i ≠ j ? k. In this paper, we first obtain formulae of determinants of power GCD matrices (S^a) and power LCM matrices [S^a] on the set S consisting of finitely many quasi-coprime divisor chains with gcd(S) ∈ S. Using these results, we then show that det(S^a)∣det(S^b), det[S^a]∣det[S^b] and det(S^a)∣det[S^b] if a∣b and S consists of finitely many quasi-coprime divisor chains with gcd(S) ∈ S. But such factorizations fail to be true if such divisor chains are not quasi-coprime.     

最大公因数矩阵的行列式

设Ｓ＝｛ｘ１，ｘ２，…，ｘｎ）是含ｎ个不同正整数的集合，（Ｓ）表示定义在Ｓ上的最大公因数矩阵，本文证明了且等号成立当且仅当Ｓ是最大公因数封闭集．     

Divisibility properties of power GCD matrices and power LCM matrices

Let a,b and n be positive integers and the set S={x₁,…,x_n} of n distinct positive integers be a divisor chain (i.e. there exists a permutation σ on {1,…,n} such that x_σ(1)|…|x_σ(n)). In this paper, we show that if a|b, then the ath power GCD matrix (S^a) having the ath power (x_i,x_j)^a of the greatest common divisor of x_i and x_j as its i,j-entry divides the bth power GCD matrix (S^b) in the ring M_n(Z) of n×n matrices over integers. We show also that if a?b and n?2, then the ath power GCD matrix (S^a) does not divide the bth power GCD matrix (S^b) in the ring M_n(Z). Similar results are also established for the power LCM matrices.     

On the divisibility of power LCM matrices by power GCD matrices

Let S = {x ₁, ..., x _n } be a set of n distinct positive integers and e ⩾ 1 an integer. Denote the n × n power GCD (resp. power LCM) matrix on S having the e-th power of the greatest common divisor (x _i , x _j ) (resp. the e-th power of the least common multiple [x _i , x _j ]) as the (i, j)-entry of the matrix by ((x _i , x _j ) ^e ) (resp. ([x _i , x _j ] ^e )). We call the set S an odd gcd closed (resp. odd lcm closed) set if every element in S is an odd number and (x _i , x _j ) ∈ S (resp. [x _i , x _j ] ∈ S) for all 1 ⩽ i, j ⩽ n. In studying the divisibility of the power LCM and power GCD matrices, Hong conjectured in 2004 that for any integer e ⩾ 1, the n × n power GCD matrix ((x _i , x _j ) ^e ) defined on an odd-gcd-closed (resp. odd-lcm-closed) set S divides the n × n power LCM matrix ([x _i , x _j ] ^e ) defined on S in the ring M _n (ℤ) of n × n matrices over integers. In this paper, we use Hong’s method developed in his previous papers [J. Algebra 218 (1999) 216–228; 281 (2004) 1–14, Acta Arith. 111 (2004), 165–177 and J. Number Theory 113 (2005), 1–9] to investigate Hong’s conjectures. We show that the conjectures of Hong are true for n ⩽ 3 but they are both not true for n ⩾ 4. Research is partially supported by Program for New Century Excellent Talents in University, by SRF for ROCS, SEM, China and by the Lady Davis Fellowship at the Technion, Israel Research is partially supported by a UGC (HK) grant 2160210 (2003/05).     

Divisibility properties of power LCM matrices by power GCD matrices on gcd-closed sets

Let e and n be positive integers and S={x₁,…,x_n} a set of n distinct positive integers. For x∈S, define . The n×n matrix whose (i,j)-entry is the eth power (x_i,x_j)^e of the greatest common divisor of x_i and x_j is called the eth power GCD matrix on S, denoted by (S^e). Similarly we can define the eth power LCM matrix [S^e]. Bourque and Ligh showed that (S)∣[S] holds in the ring of n×n matrices over the integers if S is factor closed. Hong showed that for any gcd-closed set S with |S|≤3, (S)∣[S]. Meanwhile Hong proved that there is a gcd-closed set S with max_x∈S{|G_S(x)|}=2 such that (S)?[S]. In this paper, we introduce a new method to study systematically the divisibility for the case max_x∈S{|G_S(x)|}≤2. We give a new proof of Hong’s conjecture and obtain necessary and sufficient conditions on the gcd-closed set S with max_x∈S{|G_S(x)|}=2 such that (S^e)|[S^e]. This partially solves an open question raised by Hong. Furthermore, we show that such factorization holds if S is a gcd-closed set such that each element is a prime power or the product of two distinct primes, and in particular if S is a gcd-closed set with every element less than 12.