We say that a partially ordered set $(P,\leq)$ is *interval-isomorphic* if for all $a<b \in P$ we have $P \cong [a,b]$, where $[a,b]=\{x\in P:a\leq x\leq b\}$.

Suppose $(P,\leq)$ is interval-isomorphic and there are $a,b\in P$ with $a<b$. Does this imply that $(P,\leq)$ is a lattice?