A GCD and LCM-like inequality for multiplicative lattices

Authors

  • Dan D. Anderson
  • Takashi Aoki
  • Shuzo Izumi
  • Yasuo Ohno
  • Manabu Ozaki

DOI:

https://doi.org/10.5556/j.tkjm.47.2016.1822

Keywords:

multiplicative lattice,

Abstract

Let $A_1,\ldots,A_n$ $(n\ge 2)$ 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 $$L(n)G(2)G(4)\cdots G(2\lfloor n/2 \rfloor ) \leq G(1)G(3)\cdots G(2\lceil n/2 \rceil -1).$$ In particular this holds for the lattice of ideals of a commutative ring. We also consider the relationship between $$G(n)L(2)L(4)\cdots L(2\lfloor n/2 \rfloor ) \text{ and } L(1)L(3)\cdots L(2\lceil n/2 \rceil -1)$$
and show that any inequality relationships are possible.

Author Biographies

Dan D. Anderson

Department ofMathematics, The University of Iowa, Iowa City, IA 52242, USA.

Takashi Aoki

Department ofMathematics, Kindai University, Higashi-Osaka, 577-8502, Japan.

Shuzo Izumi

Department ofMathematics, Kindai University, Higashi-Osaka, 577-8502, Japan.

Yasuo Ohno

Mathematical Institute, Tohoku University, Sendai, 980-8578, Japan.

Manabu Ozaki

Department ofMathematics,Waseda University, Shinjuku-ku Tokyo

References

D. D. Anderson, S. Izumi, Y. Ohno and M. Ozaki, GCD and LCM-like identities for ideals in commutative rings, J. Algebra Appl., 15(2016), 1650010 (12 pages).

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Published

2016-09-30

How to Cite

Anderson, D. D., Aoki, T., Izumi, S., Ohno, Y., & Ozaki, M. (2016). A GCD and LCM-like inequality for multiplicative lattices. Tamkang Journal of Mathematics, 47(3), 261-270. https://doi.org/10.5556/j.tkjm.47.2016.1822

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Section

Papers