Almost Prüfer v-Multiplication Domains and Related Domains of the Form D + D _S[Γ*]

Let D be an integral domain, S be a (saturated) multiplicative subset of D such that D {subset of with not equal to} D _S, Γ be a numerical semigroup with Γ {subset of with not equal to} ℕ₀, Γ* = Γ{set minus}{0}, X be an indeterminate over D, D + XD _S[X] = {a + Xg ∈ D _S[X]{divides}a ∈ D and g ∈ D _S[X]}, and D + D _S[Γ*] = {a + f ∈ D _S[Γ]{divides}a ∈ D and f ∈ D _S[Γ*]}; so D + D _S[Γ*] {subset of with not equal to} 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.