Let A1,…,AnA1,…,An (n≥2)(n≥2) be elements of an commutative multiplicative lattice. Let G(k)G(k) (resp., L(k)L(k)) denote the product of all the joins (resp., meets) of kk 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).
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)
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.
