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If $ (P,\leq)$ is a pre-odered set (that is, $ \leq$ is a reflexive and transitive relation) and $ x\in P$ , we set $ (\uparrow_{\leq} x) = \{p\in P: p\geq x\}$ and $ (\downarrow_{\leq} x) = \{p\in P: p\leq x\}$ . Let $ \text{NPU}(\omega)$ be the set of non-principal ultafilters on $ \omega$ .Read more