Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Non-divisibility of LCM Matrices by GCD Matrices on GCD-closed Sets   In this paper, we consider the divisibility problem of LCM matrices by GCD
matrices in the ring $M_n(\mathbb{Z})$ proposed by Hong in 2002 and in
particular a conjecture concerning the divisibility problem raised by Zhao in
2014. We present some certain gcd-closed sets on which the LCM matrix is not
divisible by the GCD matrix in the ring $M_n(\mathbb{Z})$. This could be the
first theoretical evidence that Zhao's conjecture might be true. Furthermore,
we give the necessary and sufficient conditions on the gcd-closed set $S$ with
$|S|\leq 8$ such that the GCD matrix divides the LCM matrix in the ring
$M_n(\mathbb{Z})$ and hence we partially solve Hong's problem. Finally, we
conclude with a new conjecture that can be thought as a generalization of
Zhao's conjecture.

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The categories of this paper are: q-fin
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The categories of this paper are: atom-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: adap-org
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The categories of this paper are: chem-ph
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The categories of this paper are: math-ph
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The categories of this paper are: solv-int
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The categories of this paper are: cond-mat
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The categories of this paper are: plasm-ph
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The categories of this paper are: q-alg
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The categories of this paper are: supr-con
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The categories of this paper are: patt-sol
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The categories of this paper are: eess
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The categories of this paper are: nucl-ex
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The categories of this paper are: acc-phys
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The categories of this paper are: hep-ex