A GCD and LCM-like inequality for multiplicative lattices

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

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
Section
Papers