  We use the theory of poset resolutions to construct the minimal free
resolution of an arbitrary stable monomial ideal in the polynomial ring whose
coefficients are from a field. This resolution is recovered by utilizing a
poset of Eliahou-Kervaire admissible symbols associated to a stable ideal. The
structure of the poset under consideration is quite rich and in related
analysis, we exhibit a regular CW complex which supports a minimal cellular
resolution of a stable monomial ideal.
