Consider a poset $\mathfrak{A}$. I denote $a\not\asymp b$ iff there is a non-least element $x\in\mathfrak{A}$ such that $x\le a$ and $x\le b$.

I denote $\star a = \{ x\in\mathfrak{A} \mid x\not\asymp a\}$.

Does there exist a poset $\mathfrak{A}$ such that $\star a=\star b$ implies $a=b$, but $\star a\subseteq\star b$ does not imply $a\le b$?


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