A GCD and LCM-like inequality for multiplicative lattices

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

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.

Keywords


multiplicative lattice,

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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).




DOI: http://dx.doi.org/10.5556/j.tkjm.47.2016.1822

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