@article{Zhu2022, 
author = {Guangyan Zhu and Mao Li and Xiaofan Xu},
title = {New results on the divisibility of power GCD and power LCM matrices},
year = {2022},
journal = {AIMS Mathematics},
volume = {7},
number = {10},
pages = {18239-18252},
keywords = {divisibility, power matrix, greatest-type divisor, gcd-closed set},
url = {https://www.sciopen.com/article/10.3934/math.20221003},
doi = {10.3934/math.20221003},
abstract = {Let    a  ,  b and    n be positive integers and let    S be a set consisting of    n distinct positive integers        x    1    ,  .  .  .  ,      x          n      −      1       and        x    n  . Let    (      S    a    ) (resp.    [      S    a    ]) denote the    n  ×  n matrix having    gcd  (      x    i    ,      x    j        )    a   (resp.        l    c    m    (      x    i    ,      x    j        )    a  ) as its    (  i  ,  j  )-entry. For any integer    x  ∈  S, if    (  y  <  x  ,  y      |    z      |    x        a    n    d      y  ,  z  ∈  S  )  ⇒  z  ∈  {  y  ,  x  }, then    y is called a greatest-type divisor of    x in    S. In this paper, we establish some results about the divisibility between    (      S    a    ) and    (      S    b    ), between    (      S    a    ) and    [      S    b    ] and between    [      S    a    ] and    [      S    b    ] when    a      |    b,    S is gcd closed (i.e.,    gcd  (      x    i    ,      x    j    )  ∈  S for all    1  ≤  i  ,  j  ≤  n), and        max          x      ∈      S        {      |    {  y  ∈  S  :  y    is a greatest-type divisor of    x        i    n      S  }      |    }  =  2.}
}