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 1

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    Keywords

    • GCD
    • Ideals lattice
    • LCM
    • Multiplicative lattice

    ASJC Scopus subject areas

    • Materials Science(all)
    • Metals and Alloys

    Cite this

    A GCD and LCM-like inequality for multiplicative lattices. / Anderson, Daniel D.; Aoki, Takashi; Izumi, Shuzo; Ohno, Yasuo; Ozaki, Manabu.

    In: Tamkang Journal of Mathematics, Vol. 47, No. 3, 01.09.2016, p. 261-270.

    Research output: Contribution to journalArticle

    Anderson, Daniel D. ; Aoki, Takashi ; Izumi, Shuzo ; Ohno, Yasuo ; Ozaki, Manabu. / A GCD and LCM-like inequality for multiplicative lattices. In: Tamkang Journal of Mathematics. 2016 ; Vol. 47, No. 3. pp. 261-270.
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