Meet and Join Matrices in the Poset of Exponential Divisors
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It is well-known that $(\mathbb{Z}_+,|)=(\mathbb{Z}_+,GCD,LCM)$ is a lattice, where $|$ is the usual divisibility relation and $GCD$ and $LCM$ stand for the greatest common divisor and the least common multiple of positive integers.
The number $d=\prod^r_{k=1}p^{d^{(k)}}_k$ is said to be an exponential divisor or an 𝑒-divisor of $n=\prod^r_{k=1}p^{n^{(k)}}_k(n >1)$, written as $d|_e n$, if $d^{(k)}|n^{(k)}$ for all prime divisors $p_k$ of 𝑛. It is easy to see that $(\mathbb{Z}_+\backslash\{1\},|_e)$ is a poset under the exponential divisibility relation but not a lattice, since the greatest common exponential divisor $(GCED)$ and the least common exponential multiple $(LCEM)$ do not always exist.
In this paper we embed this poset in a lattice. As an application we study the $GCED$ and $LCEM$ matrices, analogues of $GCD$ and $LCM$ matrices, which are both special cases of meet and join matrices on lattices.
Ismo Korkee^{1} ^{} Pentti Haukkanen^{1} ^{}
Volume 130, 2020
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