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.