Almost Prüferv-Multiplication Domains and Related Domains of the FormD + DS[Γ*]

Let D be an integral domain, S be a (saturated) multiplicative subset of D such that D ⊊ D S , Γ be a numerical semigroup with Γ ⊊ ℕ0, Γ* = Γ∖{0}, X be an indeterminate over D, D + XD S [X] = {a + Xg ∈ D S [X]∣a ∈ D and g ∈ D S [X]}, and D + D S [Γ*] = {a + f ∈ D S [Γ]∣a ∈ D and f ∈ D S [Γ*]}; so D + D S [Γ*] ⊊ D + XD S [X]. In this article, we study when D + D S [Γ*] is an APvMD, an AGCD-domain, an AS-domain, an AP-domain, or an AB-domain.