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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 ${\cal P}(\omega)/(\text{fin})$.

Is every homogeneous poset a lattice?

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    $\begingroup$ Any set of cardinality at least two equipped with the trivial partial order is a counterexample, since there your homogeneity condition holds vacuously. $\endgroup$
    – Tobias Fritz
    Commented May 18, 2023 at 9:18

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Counterexample. Let $$P=\{(x,i)\in\mathbb Q\times\{0,1\}:0\le x\le1,\ x\ne i\}$$ be ordered so that $$(x,i)\lt(x',i')\iff x\lt x'.$$

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