A GCD and LCM-like inequality for multiplicative lattices

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

doi:10.5556/j.tkjm.47.2016.1822

A GCD and LCM-like inequality for multiplicative lattices

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

^*Corresponding author for this work

School of Fundamental Science and Engineering

Research output: Contribution to journal › Article › peer-review

Abstract

Let A₁, . . . , A_n (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 language	English
Pages (from-to)	261-270
Number of pages	10
Journal	Tamkang Journal of Mathematics
Volume	47
Issue number	3
DOIs	https://doi.org/10.5556/j.tkjm.47.2016.1822
Publication status	Published - 2016 Sept

Keywords

GCD
Ideals lattice
LCM
Multiplicative lattice

ASJC Scopus subject areas

Mathematics(all)
Applied Mathematics

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10.5556/j.tkjm.47.2016.1822

Cite this

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title = "A GCD and LCM-like inequality for multiplicative lattices",

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.",

keywords = "GCD, Ideals lattice, LCM, Multiplicative lattice",

author = "Anderson, {Daniel D.} and Takashi Aoki and Shuzo Izumi and Yasuo Ohno and Manabu Ozaki",

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doi = "10.5556/j.tkjm.47.2016.1822",

language = "English",

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journal = "Tamkang Journal of Mathematics",

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TY - JOUR

T1 - A GCD and LCM-like inequality for multiplicative lattices

AU - Anderson, Daniel D.

AU - Aoki, Takashi

AU - Izumi, Shuzo

AU - Ohno, Yasuo

AU - Ozaki, Manabu

PY - 2016/9

Y1 - 2016/9

N2 - 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.

AB - 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.

KW - GCD

KW - Ideals lattice

KW - LCM

KW - Multiplicative lattice

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JO - Tamkang Journal of Mathematics

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ER -

A GCD and LCM-like inequality for multiplicative lattices

Abstract

Keywords

ASJC Scopus subject areas

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