Divisibilty Properties of Gcd Ve Lcm Matrices

Let a, b and n be positive integers and let S = {x1, x2, . . . , xn} be a set of distinct positive integers. The n × n matrix (Sf ) = (f ((xi, xj))), having f evaluated at the greatest common divisor (xi, xj) of xi and xj as its ij−entry, is called the GCD matrix associated with f on the set S. Similarly, the n × n matrix [Sf ] = (f ([xi, xj ])) is called the LCM matrix associated with f on S. If f = I, the identity function on Z, we have the classical GCD matrix (S) and the classical LCM matrix [S]. If f = N, the power function, we have the a−th power GCD matrix (S) and a−th the power LCM matrix [S]. Let f be an integer valued arithmetical function. It is said that the matrix (Sf ) divides the matrix [Sf ] in M(n,Z) and denoted by (Sf ) | [Sf ] if there exists an n×n matrix B ∈M(n,Z) such that [Sf ] = (Sf )B. In [1], Bourque and Ligh showed that if S is factor closed then (S)|[S]. The set S is said to be factor closed if it contains every positive divisor of x for any x ∈ S. Hong [4] showed that such factorization is no longer true in general if S is gcd-closed. The set S is said to be gcd-closed if (xi, xj) ∈ S for all 1 ≤ i, j ≤ n. In this frame, many results on divisibility among GCD, LCM and related matrices are published in the literature. In this talk, we summarize these results and open problems.