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      Permanent link:
      https://www.ias.ac.in/article/fulltext/pmsc/119/03/0319-0332

    • Keywords

       

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

    • Abstract

       

      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.

    • Author Affiliations

       

      Ismo Korkee1 Pentti Haukkanen1

      1. Department of Mathematics, Statistics and Philosophy, University of Tampere, FI-33014, Finland
    • Dates

       
  • Proceedings – Mathematical Sciences | News

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