Hi
I'm looking for an elementary substructure of a natural numbers ultrapower with a free ultrafilter over a numerable set also must not be isomorphism between the elementary substructure and any ultrapower with an lower ultrafilter in Rudin-Keisler order.
Sorry for my awful and unpleasant english
Thanks
This depends on how which fundamental operations and relations you want on the natural numbers to begin with. For instance, if you have at most countably many operations on $\mathbb{N}$, then you can easily get many elementary substructures. Take note that any ultrapower $\mathbb{N}^{\mathcal{U}}$ by a non-principal ultrafilter $\mathcal{U}$ on a countable set has cardinality continuum. Take for instance $(\lfloor nr\rfloor)_{n}/\mathcal{U}$ as $r$ ranges over the positive real numbers. This set will have cardinality continuum. Therefore by the Lowenheim-Skolem theorem, the ultrapower $\mathbb{N}^{\mathcal{U}}$ has many countable elementary substructures that properly extend $\mathbb{N}$.
On the other hand, if you let all operations on $\mathbb{N}$ be fundamental operations and all relations be fundamental operations, then it is possible that $\mathbb{N}^{\mathcal{U}}$ has no non-trivial elementary substructures, and we shall be able to characterize the elementary substructures of $\mathbb{N}$. Let $\Omega(\mathbb{N})$ be the structure with underlying set $\mathbb{N}$ and where each operation is a fundamental operation, each constant is a fundamental constant, and each relation is a fundamental relation. We first take note if $\mathcal{A}$ is a model of $\mathrm{Th}(\Omega(\mathbb{N}))$, then every substructure of $\mathcal{A}$ is an elementary substructure.
If $f:\mathbb{N}\rightarrow\mathbb{N}$, then let $\Pi(f)$ be the partition of $\mathbb{N}$ where $x=y\,(\Pi(f))$ if and only if $f(x)=f(y)$. Let $\mathcal{U}$ be a non-principal ultrafilter over $\mathbb{N}$. Let $\mathrm{Sub}(\Omega(A)^{\mathcal{U}})$ denote the lattice of elementary substructres of $\Omega(A)^{\mathcal{U}}$. Let $\Pi(\mathbb{N})$ denote the lattice of partitions of $\mathbb{N}$. In particular, $\Pi(\mathbb{N})$ is a meet-semilattice. Let $\Theta_{\mathcal{U}}$ be the congruence on the meet-semilattice $\Pi(\mathbb{N})$ where $(P,Q)\in\Theta_{\mathcal{U}}$ if and only if there is some $R\in\mathcal{U}$ where $\{A\cap R|A\in P\}\setminus\{\emptyset\}=\{B\cap R|B\in Q\}\setminus\{\emptyset\}$.` Then let $\Pi(\mathbb{N})/\mathcal{U}$ denote the quotient meet-semilattice $\Pi(\mathbb{N})/\Theta_{\mathcal{U}}$. Let $\mathrm{Fi}(\Pi(\mathbb{N})/\mathcal{U})$ denote the lattice of filters on the meet-semilattice $\Pi(\mathbb{N})/\mathcal{U}$. Define a map $L:\mathrm{Fi}(\Pi(\mathbb{N}))\rightarrow\mathrm{Sub} (\Omega(\mathbb{N})^{\mathcal{U}})$ by letting $L(Z)=\{[f]\in\Omega(\mathbb{N})^{\mathcal{U}}:\Pi(f)/\Theta_{\mathcal{U}}\in Z\}$. Then $L$ is an order reversing bijection, and this bijection restricts to a bijection between the principal filters on the meet-semilattice $L$ and the finitely generated substructures of $\Omega(\mathbb{N})^{\mathcal{U}}$. The finitely generated substructures of $\Omega(\mathbb{N})^{\mathcal{U}}$ are generated by a single element, and the finitely generated substructures of $\Omega(\mathbb{N})^{\mathcal{U}}$ correspond to the structures obtained by going down from $\mathcal{U}$ on the Rudin-Keisler preordering. Every elementary substructure of $\Omega(\mathbb{N})^{\mathcal{U}}$ is obtained by going down from $\mathcal{U}$ on the Rudin-Kiesler preordering if and only if the meet-semilattice $\Pi(\mathbb{N})/\mathcal{U}$ satisfies the descending chain condition. This condition is equivalent to the following restatement: if $(P_{n})_{n}$ is a sequence of partitions with $P_{n}\preceq P_{n-1}$ for all $n$, then there is some $N$ where if $n\geq N$, then there is some $R_{n}\in\mathcal{U}$ where if $A\in P_{n+1}$, then there is some $B\in P_{n}$ with $A\cap R_{n}=B\cap R_{n}$ (do these kinds of ultrafilters have a name?). I should note that the one-to-one correspondence between $\mathrm{Fi}(\Pi(\mathbb{N})/\mathcal{U})$ and $\mathrm{Sub}(\Omega(\mathbb{N})^{\mathcal{U}})$ can be generalized to different kinds of ultrapower constructions and even reduced power constructions including Boolean ultrapowers, limit ultrapowers, limit reduced powers, etc. As a side note, I personally have found this characterization of subalgebras useful for studying things such as the Boolean ultrapower and determining when a Boolean ultrapower is an ultrapower.
For example, an ultrafilter $\mathcal{U}$ on $\mathbb{N}$ is selective if and only if the lattice $\Pi(\mathbb{N})/\mathcal{U}$ has only two elements, and selective ultrafilters are precisely the ultrafilters such that $\Omega(A)^{\mathcal{U}}$ has no elementary substructures besides $\Omega(A)$ and $\Omega(A)^{\mathcal{U}}$. Selective ultrafilters may be constructed using the continuum hypothesis.
On the other hand, if $(P_{n})_{n}$ is a descending sequence of partitions of $\mathbb{N}$, where if $A\in P_{n}$, then $\{B\in P_{n+1}|B\subseteq A\}$ is infinite, then it is easily shown that there is an ultrafilter $\mathcal{U}$ such that $P_{n+1}/\Theta_{\mathcal{U}}\neq P_{n}/\Theta_{\mathcal{U}}$ for all $n$.
