Let $L=\{P_0,P_1,P_2\}$ be a first order language, and let $$T=\bigg\{\Big(\forall x\ P_i(x)\Big)\vee \Big(\forall x\ \neg P_i(x)\Big):i \in \{0,1,2\}\bigg\}\\ \qquad\qquad\qquad\qquad\qquad\bigcup\bigg\{\forall x\ P_0(x)\implies P_1(x), \forall x\ P_1(x)\implies P_2(x) \bigg\}\bigcup T_\infty,$$
where $T_\infty$ is just the theory of an infinite model.
I.e. any of the $P_i$'s is either empty or the whole set, and $P_0\subseteq P_1\subseteq P_2$.
I believe (correct me if wrong) that there are 4 countable non-isomorphic models of $T$:
$\mathfrak{A}_0=(A_0;\emptyset,\emptyset,\emptyset)$,
$\mathfrak{A}_1=(A_1;\emptyset,\emptyset,A_1)$,
$\mathfrak{A}_2=(A_2;\emptyset,A_2,A_2)$, and
$\mathfrak{A}_3=(A_3;A_3,A_3,A_3)$.
Now, my questions are:
Are any of those models the prime model of $T$? Are any of them saturated? $\omega$-homogeneous? Atomic?
I would think none of them is a prime model, since in $\mathfrak{A}_0$ (the only one that could be a prime model) the sentence $\exists x P_1(x)$ fails, but it's true in $\mathfrak{A_3}$.
Somehow my intuition tells me they are $\omega$-homogeneous. Somehow one would take advantage of the fact that any $P_i$ is either everything or nothing, hence any bijection would be elementary, right?
I'm not too sure how to go about with saturation, nor atomicity (is that the word?), though. I figure arbitrary types over finite parameteres could get really complicated. And I don't have enough intuition about atomic models to really know what to do.
Any ideas?