  In this paper we investigate the concept of radical factorization with
respect to finitary ideal systems of cancellative monoids. We present new
characterizations for r-almost Dedekind r-SP-monoids and provide specific
descriptions of t-almost Dedekind t-SP-monoids and w-SP-monoids. We show that a
monoid is a w-SP-monoid if and only if the radical of every nontrivial
principal ideal is t-invertible. We characterize when the monoid ring is a
w-SP-domain and describe when the *-Nagata ring is an SP-domain for a star
operation * of finite type.
