On Characteristic Poset and Stanley Decomposition

Let J ⊂ I be two monomial ideals such that I/J is Cohen Macaulay. By associating a finite posets PI/Jg$P_{I/J}^g$ to I/J, we show that if I/J is a Stanley ideal then I/J˜$\widetilde{I/J}$ is also a Stanley ideal, where I/J˜$\widetilde{I/J}$ is the polarization of I/J. We also give relations between sdepth and fdepth of I/J and I/J˜$\widetilde{I/J}$