A GCD and LCM-like inequality for multiplicative lattices

Daniel D. Anderson, Takashi Aoki, Shuzo Izumi, Yasuo Ohno, Manabu Ozaki

Research output: Contribution to journalArticle

Abstract

Let A1, . . . , An (n ≥ 2) be elements of an commutative multiplicative lattice. Let G(k) (resp., L(k)) denote the product of all the joins (resp., meets) of k of the elements. Then we show that L(n)G(2)G(4) ···G(2[n/2]) ≤ G(1)G(3) ···G(2[n/2]-1). In particular this holds for the lattice of ideals of a commutative ring. We also consider the relationship between G(n)L(2)L(4) ···L(2[n/2]) and L(1)L(3) ···L(2[n/2]-1) and show that any inequality relationships are possible.

Original languageEnglish
Pages (from-to)261-270
Number of pages10
JournalTamkang Journal of Mathematics
Volume47
Issue number3
DOIs
Publication statusPublished - 2016 Sep

Keywords

  • GCD
  • Ideals lattice
  • LCM
  • Multiplicative lattice

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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