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

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A GCD and LCM-like inequality for multiplicative lattices

Journal article

Open access

Peer reviewed

A GCD and LCM-like inequality for multiplicative lattices

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

Tamkang Journal of Mathematics, Vol.47(3), pp.261-270

2016

DOI: 10.5556/j.tkjm.47.2016.1822

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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.

GCD

Ideals lattice

LCM

Multiplicative lattice

Details

Title: Subtitle: A GCD and LCM-like inequality for multiplicative lattices
Creators: Daniel D Anderson - University of Iowa
Takashi Aoki - Kindai University
Shuzo Izumi - Kindai University
Yasuo Ohno - Tohoku University
Manabu Ozaki - School of Fundamental Science and Engineering
Resource Type: Journal article
Publication Details: Tamkang Journal of Mathematics, Vol.47(3), pp.261-270
DOI: 10.5556/j.tkjm.47.2016.1822
ISSN: 0049-2930
Publisher: Tamkang University
Language: English
Date published: 2016
Academic Unit: Mathematics
Record Identifier: 9984230626702771

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