A "Berning" Question of Krebs and van der Zypen
Keywords:
exponentiation, Priestley duality, Priestley spaceAbstract
Given a poset $\left(P, \leq_P\right)$, let $\Phi(P)$ be the binary relation $\leq_P$ viewed as a subposet of $P \times P$. Michel Krebs and Dominic van der Zypen [Topology Proc. 31 (2007), no. 2, 583-591] asked if there was a connected poset $Y$ that was not a lattice such that $\Phi(Y) \cong Y$. In the present article, such an example is given.
Consider Priestley spaces $X$ that are order-homeomorphic to $X \times \mathbf{2}$, where $\mathbf{2}$ is viewed as a Priestley space. Krebs and van der Zypen asked if the Priestley space $\mathbf{2}^{\mathcal{N}_0}$ embeds in such an $X$. If not, they asked if such an $X$ can be countably infinite.
Krebs and Jürg Schmid [J. Log. Algebr. Program. 76 (2008), no. 2, 198-208] proved that if $X \neq \emptyset$, then there is a surjective morphism from $X$ onto $\mathbf{2}^{\mathrm{N}_0}$. Using this, we show that there is an order-embedding from $\mathbf{2}^{\mathrm{N}_0}$ to $X$ if $X \neq \emptyset$.
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