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?



Are non-trivial interval-isomorphic posets lattices?
No. Let $P^-=\mathbb Q\times\{0,1\}$ with partial order defined by
$$\langle x,a\rangle\le\langle y,b\rangle\iff x<y\lor(x=y\land a=b),$$
and let $P=P^-\cup\{-\infty,+\infty\}$ with $-\infty<\langle x,a\rangle<+\infty$.


