A poset $(P,\leq)$ is *homogeneous* if $P\cong [a,b]$ for all $a,b\in P$ with $a<b$ (where $[a,b] := \{x\in P: a\leq x\leq b\}$).


Examples of homogeneous posets include $[0,1]$, $[0,1]\cap \mathbb{Q}$, and [the Boolean algebra ${\cal P}(\omega)/(\text{fin})$](https://mathoverflow.net/questions/446932/is-cal-p-omega-textfin-a-fractal-poset).


Is every homogeneous poset a [lattice](https://en.wikipedia.org/wiki/Lattice_(order))?



Is every homogeneous poset a lattice?
**Counterexample.** Let
$$P=\{(x,i)\in\mathbb Q\times\{0,1\}:0\le x\le1,\ x\ne i\}$$
be ordered so that
$$(x,i)\lt(x',i')\iff x\lt x'.$$


