### Abstract

Let A_{1}, . . . , 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 | |

Publication status | Published - 2016 Sep 1 |

### Fingerprint

### Keywords

- GCD
- Ideals lattice
- LCM
- Multiplicative lattice

### ASJC Scopus subject areas

- Materials Science(all)
- Metals and Alloys

### Cite this

*Tamkang Journal of Mathematics*,

*47*(3), 261-270. https://doi.org/10.5556/j.tkjm.47.2016.1822

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

Research output: Contribution to journal › Article

*Tamkang Journal of Mathematics*, vol. 47, no. 3, pp. 261-270. https://doi.org/10.5556/j.tkjm.47.2016.1822

}

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 - 2016/9/1

Y1 - 2016/9/1

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

VL - 47

SP - 261

EP - 270

JO - Tamkang Journal of Mathematics

JF - Tamkang Journal of Mathematics

SN - 0049-2930

IS - 3

ER -