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Priestley Duality assigns to every bounded distributive lattice $L$ a compact totally order-disconnected topological space $P(L)$, also called a Priestley space.

A poset $(P,\leq)$ is called (Priestley-) representable if there is a topology $\tau$ on $P$ such that $(P,\tau,\leq)$ is a Priestley space. Moreover, $(P,\leq)$ is called

  1. uniquely representable if whenever there are topologies $\tau_1, \tau_2$ on $P$ such that $(P,\tau_i,\leq)$ is a Priestley space for $i=1,2$, we have $\tau_1=\tau_2$;
  2. weakly uniquely representable if whenever there are topologies $\tau_1, \tau_2$ on $P$ such that $(P,\tau_i,\leq)$ is a Priestley space for $i=1,2$, the spaces $(P,\tau_1,\leq)$ and $(P,\tau_2,\leq)$ are order-homeomorphic (i.e. there is an automorphism $\varphi:(P,\tau_1) \to (P,\tau_2)$ that is also an order-isomorphism).

Question. Is there a weakly uniquely representable poset $(P,\leq)$ that is not uniquely representable?

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