This is Chapter 3, Problem 10b from Stanley's Enumerative Combinatorics.
Let $P$ and $Q$ be posets. If $P$ has a $\hat{0}$ and $Int(P) \cong Int(Q)$, show $\exists$ posets $ A, B$ so that $P \cong A \times B^*$ and $Q\cong A \times B$.
Following Stanley's hint, I let $f$ be an isomorphism $Int(P)\to Int(Q)$, and let $s ∈ Q$ be such that $f[0, 0]=[s, s]$. Then I defined $A = \{q \in Q: q \ge s\}$, $B=\{q \in Q: q \le s\}$. I was able to show that $P \cong A \times B^*$ via the following map:
Suppose $f[0, a]=[b_a, c_a] \in Int(Q).$ Then we define $\phi(a)=(c_a, b_a) \in A \times B^*$.
However, I'm unsure how to show that $Q \cong A \times B$. I attempted to define a map $\eta: A \times B \to Q$ via:
Let $a \in A, b \in B.$ Then $b \le s \le a$. Suppose $f^{-1}[b,a]=[0, h].$ Then $\eta(a, b):= f[h, h]$.
I'm struggling to show that $\eta$ is an order-preserving map between posets, and I'm starting to question whether $\eta$ is the right map to use. How can I find an isomorphism $A \times B \to Q$?