Meet and Join Matrices in the Poset of Exponential Divisors

Proceedings Mathematical Sciences

Pentti Haukkanen¹  Ismo Korkee² 

[1] $$;Department of Mathematics, Statistics and Philosophy, University of Tampere, FI-0, Finland$$

关键词: Exponential divisor;    lattice;    meet matrix;    join matrix;    greatest common divisor matrix;    least common multiple matrix.;   

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

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 $(mathbb{Z}_+ackslash{1},|_e)$ 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.

【 授权许可】

Unknown