  We give a combinatorial description for the weak order on the hyperoctahedral
group. This characterization is then used to analyze the order-theoretic
properties of the shifted products of hyperoctahedral groups. It is shown that
each shifted product is a disjoint union of some intervals, which can be convex
embedded into a hyperoctahedral group. As an application, we investigate the
monomial basis for the Hopf algebra $\mathfrak{H}Sym$ of signed permutations,
related to the fundamental basis via M\"obius inversion on the weak order on
hyperoctahedral groups. It turns out that the image of a monomial basis element
under the descent map from $\mathfrak{H}Sym$ to the algebra of type $B$
quasi-symmetric functions is either zero or a monomial quasi-symmetric function
of type $B$.
