Divisibility Of Smith Matrices On GCD-closed Sets

Arithmetic functions and integer sets are two important research objects in num-ber theory.The divisibility of matrices associated with arithmetic functions is one of the most important research topics in combinatorial number theory.It’s research methods and results have important applications in many problems of combinatorial number theory,matrix theory and partial order theory.This paper mainly studies the arithmetic properties of Smith matrices on GCD-closed sets,with a focus on the divisibility between matrices associated with arithmetic functions.Let n be a positive integer,S={x₁,...,x_n}be the set of positive integers,（x_i,x_j）be the greatest common divisor of x_iand x_j,and[x_i,x_j]be the least common multiple of x_iand x_j.The set S is said to be a GCD-closed set if the greatest common divisor of any two elements of S is in S.Let x,y∈S with y<x,if y|x,the conditions y|z|x and d∈S imply that d∈{x,y},we say that y is a greatest-type divisor of x in S.For x∈S,G_S（x）stands for the set of all greatest-type divisors of x in S.For a given arithmetic function f and a set S of positive integers,（f（S））=（f（x_i,x_j））and（f[S]）=（f[x_i,x_j]）are usually called Smith matrices.For integer matrices A and B,if there exists an integer matrix C such that B=AC or B=CA,then the matrix A divides the matrix B.In this thesis,we focus on characterizing the GCD-closed set S and arithmetic function f such that in the ring of integer matrices M_n（Z）,the matrix（f（S））divide the matrix（f[S]）.First,we study the divisibility of matrices associated with a class of arithmetic functions on GCD-closed set S with max（?）{|G_S（x）|}=1.By analyzing the divisibil-ity structure of S,we obtain a sufficient and necessary condition for（f（S））divides（f[S]）,which generalizes the results of Li and Tan in 2011.Then we give a sufficient condition for（f^a（S））|（f^b[S]）,which generalizes the result of Feng et al.in 2022.Second,we consider the divisibility of Smith matrices on a GCD-closed set S with max（?）{|G_S（x）|}=2.An element x is said to satisfy the condition C if[y,z]=x and（y,z）∈G_S（y）∩G_S（z）for any y,z∈G_S（x）with y≠z.A set S is said to satisfy the condition C if each elements x∈S with|G_S（x）|≥2 satisfy the condition C.For a given class of arithmetic functions,we prove that if S satisfies the condition C,then（f（S））divides（f[S]）.Last,we define a new class of arithmetic functions and prove that（f（S））divides（f[S]）if and only if S satisfies the condition C.This generalizes the result of Feng,Hong and Zhao in 2009,and partially answers the question proposed by Feng et al.in 2022.