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

^*この研究の対応する著者

基幹理工学部

研究成果: Article › 査読

抄録

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.

本文言語	English
ページ（範囲）	261-270
ページ数	10
ジャーナル	Tamkang Journal of Mathematics
巻	47
号	3
DOI	https://doi.org/10.5556/j.tkjm.47.2016.1822
出版ステータス	Published - 2016 9月

ASJC Scopus subject areas

数学 (全般)
応用数学

文献へのアクセス

10.5556/j.tkjm.47.2016.1822

他のファイルとリンク

引用スタイル

@article{49d527ab880b426f87e75f7ed1323bbb,

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",

year = "2016",

month = sep,

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

language = "English",

volume = "47",

pages = "261--270",

journal = "Tamkang Journal of Mathematics",

issn = "0049-2930",

publisher = "Tamkang University",

number = "3",

}

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

UR - http://www.scopus.com/inward/record.url?scp=84986570819&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84986570819&partnerID=8YFLogxK

U2 - 10.5556/j.tkjm.47.2016.1822

DO - 10.5556/j.tkjm.47.2016.1822

M3 - Article

AN - SCOPUS:84986570819

SN - 0049-2930

VL - 47

SP - 261

EP - 270

JO - Tamkang Journal of Mathematics

JF - Tamkang Journal of Mathematics

IS - 3

ER -

A GCD and LCM-like inequality for multiplicative lattices

抄録

ASJC Scopus subject areas

文献へのアクセス

他のファイルとリンク

フィンガープリント

引用スタイル