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Suppose that $\mathcal{Z}$ is a filter on a set $X$. Let $\Pi(X)$ denote the lattice of all partitions of the set $X$. Then $(\Pi(X),\wedge)$ is a meet-semilattice where $P\wedge Q=\{R\cap S|R\in P,S\in Q\}\setminus\{\emptyset\}$. Let $\simeq_{\mathcal{Z}}$ be the equivalence relation on $\Pi(X)$ where $P\simeq_{\mathcal{Z}}Q$ if and only if there is some $S\in\mathcal{Z}$ where $\{S\cap R|R\in P\}\setminus\{\emptyset\}=\{S\cap R|R\in Q\}\setminus\{\emptyset\}$. Then $\simeq_{\mathcal{Z}}$ is a congruence on the meet-semilattice $(\Pi(X),\wedge)$. Let $P_{\mathcal{Z}}$ be the quotient meet-semilattice $(\Pi(X)/\simeq_{\mathcal{Z}},\wedge).$ For ultrafilters, the meaning behind the meet-semilattice $(\Pi(X)/\simeq_{\mathcal{U}},\wedge)$ is that we regard two partitions $P,Q$ as being equal if for all structures $\mathcal{A}$, their corresponding $``$sub-ultrapowers$"$ are equal i.e. $$\{[f]\in\mathcal{A}^{\mathcal{U}}|P\preceq\Pi(f)\}=\{[f]\in\mathcal{A}^{\mathcal{U}}|Q\preceq\Pi(f)\}$$ where $\Pi(f)=\{f^{-1}[\{a\}]|a\in\mathcal{A}\}\setminus\{\emptyset\}$ and $\preceq$ is the refinement ordering on the lattice of partitions. In other words, $\Pi(X)/\simeq_{\mathcal{U}}$ is up to isomorphism the meet-semilattice of all the subultrapowers of a structure ordered under reverse inclusion. If $\mathcal{Z}$ is simply a filter, then $P_{\mathcal{Z}}$ is up-to-isomorphism the meet-semilattice of all sub-reduced powers ordered under reverse inclusion.

  1. Does there exist a model of ZFC where the ultrafilters $\mathcal{U}$ and $\mathcal{V}$ are Rudin-Kiesler equivalent if and only if $P_{\mathcal{U}}$ is isomorphic to $P_{\mathcal{V}}$?

  2. More generally, does there exist a model of ZFC where two filters $\mathcal{Z}_{1},\mathcal{Z}_{2}$ on sets are isomorphic if and only if the meet-semilattices $P_{\mathcal{Z}_{1}}$ and $P_{\mathcal{Z}_{2}}$ are isomorphic?

This property in question 1 does not hold for many models of ZFC. In particular, if $\mathcal{U},\mathcal{V}$ are two non-Rudin Keisler equivalent selective ultrafilters, then $P_{\mathcal{U}}\simeq P_{\mathcal{V}}\simeq\{0,1\}$. In particular, under CH or more generally MA, there are $2^{\mathfrak{c}}$ non-Rudin Keisler equivalent ultrafilters on $\omega$. Furthermore, if there are two distinct measurable cardinals, then the normal ultrafilters $\mathcal{U}_{1},\mathcal{U}_{2}$ on those measurable cardinals are not Rudin Keisler equivalent, but $P_{\mathcal{U}_{1}}\simeq P_{\mathcal{U}_{2}}\simeq\{0,1\}$.

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