Filtered Products

Prove that if $D$ is a filter over $I$, then $I \in D$.

Let $X \subseteq I$. Prove that $\left\{ Y \mid Y \subseteq I \text{ and } X \subseteq Y\right\}$ is a filter over $I$.

Let $D$ be a filter over $I$ and $J \in D$. Show that $D_{1} = \left\{ X \cap J \mid X \in D\right\}$ is a filter over $J$, and also that if $D$ is a nonprincipal filter, then $D_{1}$ is also a nonprincipal filter.

Prove that if some finite set belongs to a filter, then this filter is principal.

Prove that every nonprincipal ultrafilter contains all sets having finite complements.

Prove that a filter $D$ over $I$ is an ultrafilter if and only if $D$ is a maximal element in the set of all filters over $I$, ordered by inclusion.

Prove that in the set $\Phi$ of all filters over $I$, ordered by inclusion, $\left\{ I\right\}$ is the least element. Show also that if $\left|I\right| \geq 2$, then $\Phi$ has no greatest element.

Prove that, for a filter over $I$ containing the set $S \subseteq \mathcal{P}(I)$ to exist, it is necessary and sufficient that the intersection of any finite number of elements of $S$ be nonempty.

Let $I$ be an infinite set of cardinality $\alpha$ and

Prove that $\Phi$ is a filter over $I$.

Prove that a system $\Psi$ of subsets of a set $I$ is contained in some Frechet filter if and only if every intersection of finitely many sets of the system $\Psi$ has cardinality equal to the cardinality of the set $I$.

Show that every nonprincipal ultrafilter over a countable set is a Frechet filter.

Let $F$ be a filter over $I$, $A \subseteq I$, and $F_{A} = \left\{ X \cap A \mid X \in F\right\}$. Prove that, for $F_{A}$ to be a filter over $A$, it is necessary and sufficient that $X \cap A \neq \emptyset$ for every $X \in F$.

Let $F$ be an ultrafilter over $I$, $A \subseteq I$, and $F_{A} = \left\{ X \cap A \mid X \in F\right\}$. Prove that, for $F_{A}$ to be an ultrafilter over $A$, it is necessary and sufficient that $A$ belong to $F$.

Show that every filter can be extended to an ultrafilter.

Prove that every filter is the intersection of all ultrafilters containing it.

Let $F$ and $G$ be filters over $I$. Prove that

Prove that if the union of a finite sequence $\left\{ A_{i}\right\}_{i \leq n}$ of subsets of a set $I$ belongs to an ultrafilter $F$, then at least one of the sets $A_{i}$ belongs to $F$.

Let $F$ be an ultrafilter and $G_{1}, \ldots , G_{n}$ filters over $I$. Prove that if $F \supseteq G_{1} \cap \ldots \cap G_{n}$, then there exists $i$ ($1 \leq i \leq n$) such that $F \supseteq G_{i}$.

Let the set $J$ be infinite. Construct an ultrafilter $F$ and a family of ultrafilters $\left\{ G_{j}\right\}_{j \in J}$ such that $F \supseteq \bigcap_{j \in J} G_{j}$, but $F$ does not contain $G_{j}$ for any $j$.

Prove that $\bigcap_{X \in F} X$, where $F$ is an ultrafilter, contains at most one point.

Let $D$ be a filter over $I$ and $\sim$ some equivalence relation on $I$. Let $D^{\sim } = \left\{ B \mid \exists A (A \in D \& B = \left\{ [x]_{\sim } \mid x \in A\right\} )\right\}$. Prove that:

Prove that a filter $D$ over $I$ is countably complete if and only if there is no decreasing sequence $X_{0} \supset X_{1} \supset X_{2} \supset \ldots$ of elements $X_{i} \in D$ such that $\bigcap_{i \in \mathbb {N}} X_{i} = \emptyset$.

Prove that $\sim_{D}$ is an equivalence relation on $\prod_{i \in I} M_{i}$.

Prove that if $f_{1} \sim_{D} g_{1}, \ldots , f_{n} \sim_{D} g_{n}$, then:

Prove that for every $I$,

where $D = \left\{ I\right\}$.

Prove that the canonical mapping $\varphi : \prod_{i \in I} \mathfrak {M}_{i} \to \prod_{i \in I} \mathfrak {M}_{i} / D$, where $\varphi (f) = f / D$, is a homomorphism of $\prod_{i \in I} \mathfrak {M}_{i}$ onto $\prod_{i \in I} \mathfrak {M}_{i} / D$. Show that, in general, this homomorphism is not a strong homomorphism.

Prove that if a sentence containing neither $\neg$ nor $\supset$ is true on the direct product $\prod_{i \in I} \mathfrak {M}_{i}$, then it is true on the filtered product $\prod_{i \in I} \mathfrak {M}_{i} / D$ with respect to any filter $D$.

Prove that if $D, D_{1}$ are filters over $I$ and $D \subseteq D_{1}$, then the mapping $\varphi$ defined by $\varphi (f / D) = f / D_{1}$ is a homomorphism of $\prod_{i \in I} \mathfrak {M}_{i} / D$ onto $\prod_{i \in I} \mathfrak {M}_{i} / D_{1}$.

Let $J \in D$, where $D$ is a filter over $I$. Show that

where $D_{J}$ is the filter formed by the intersections of $J$ with the sets of the filter $D$. In particular, if $D$ is a principal filter consisting of subsets of the set $J$, then

Let $I$ be finite and $D$ a filter over $I$. Prove that $\prod_{i \in I} \mathfrak {M}_{i} / D$ is isomorphic to the direct product of some of the systems $\mathfrak {M}_{i}$.

Let $\left\{ I_{k} \mid k \in K\right\}$ be a partition of $I$, let filters $D_{k}$ be given over $I_{k}$, and let a filter $D^{*}$ be given over $K$. Show that

is a filter over $I$ and, for all $\mathfrak {M}_{i}$ ($i \in I$),

(the associative law for filtered products).

Prove that if the intersection $J$ of all sets of a filter $D$ over $I$ is nonempty and does not belong to $D$, then

where $J' = I \backslash J$ and $D_{J'}$ is the filter over $J'$ formed by the intersections of $J'$ with all sets of the filter $D$.

Let $\varphi_{i}: M_{i} \to N_{i}$ be homomorphisms, and let $\varphi (f / D) = (\varphi '(f)) / D$, where $(\varphi '(f))(j) = \varphi_{j}(f(j))$ for $f \in \prod_{i \in I} M_{i}$, $j \in J$. Prove that then $\varphi$ is a homomorphism of $\prod_{i \in I} \mathfrak {M}_{i} / D$ into $\prod_{i \in I} \mathfrak {N}_{i} / D$. Prove that if the $\varphi_{i}$ are strong homomorphisms (isomorphisms), then so is $\varphi$.

Let $\varphi$ be a one-to-one correspondence between $I$ and $J$, let $D$ be a filter over $I$, and $D_{1} = \left\{ \varphi (X) \mid X \in D\right\}$. Prove that

Let $\left\{ C_{i} \mid i \in I\right\}$ be a family of sets, and let an equivalence relation $\sim$ be given on $I$ such that $i \sim j \Rightarrow C_{i} = C_{j}$. Prove that for every ultrafilter $D$ over $I$ there exists an isomorphic embedding of $\prod_{\alpha \in I / {\sim }} C_{\alpha } / D^{\sim }$ into $\prod_{i \in I} C_{i} / D$, where $D^{\sim }$ is constructed as in Problem II.8.21, and $C_{\alpha } = C_{i}$ for $\alpha \in I / {\sim }$ and $i \in \alpha$.

Let $\left\{ I_{k}\right\}_{k \in K}$ be the family of all finite nonempty subsets of $I$, and let $\left\{ \mathfrak {M}_{i}\right\}_{i \in I}$ be a family of algebraic systems of signature $\sigma$. Prove that there exists a filter $D$ over $K$ such that for every filter $D_{1} \supseteq D$ over $K$ there exists an isomorphic embedding of $\prod_{i \in I} \mathfrak {M}_{i}$ into $\prod_{k \in K} \left( \prod_{i \in I_{k}} \mathfrak {M}_{i} \right) \Big/ D_{1}$.

Prove that if $A$ and $B$ are conditionally filterable with respect to the filter $D$ over $I$, then $(A \& B)$, $\exists x \, B$, $\forall x \, A$ are also conditionally filterable with respect to $D$.

Prove that if $A$ and $B$ are filterable with respect to the filter $D$ over $I$, then $(A \& B)$, $\exists x \, A$ are also filterable with respect to $D$.

Prove that atomic formulas are filterable with respect to every filter.

Let $A$ be filterable with respect to the ultrafilter $D$. Prove that $\neg A$ is filterable with respect to $D$.

Prove that every formula is filterable with respect to every ultrafilter (os's theorem).

Let $A$ be filterable with respect to the filter $D$ over $I$, and let $B$ be conditionally filterable with respect to $D$. Prove that $(A \supset B)$ and $\neg A$ are conditionally filterable with respect to $D$.

Call a formula Horn if it is obtained from formulas of the form $\neg A$, $(A \supset B)$, $B$, where $A$ is a conjunction of atomic formulas and $B$ is an atomic formula, by means of the operations of conjunction and adjoining quantifiers. Prove that Horn formulas are conditionally filterable with respect to every filter.

Prove that the ultraproduct of linearly ordered sets is linearly ordered.

Prove that the filtered product of preordered sets is preordered.

Prove that the filtered product of partially ordered sets is partially ordered.

Prove that the filtered product of groups is a group.

Let a finite system $\mathfrak {M}$ contain $n$ elements. Prove that every ultrapower of $\mathfrak {M}$ also contains $n$ elements.

Show that if the cardinalities of all the factors do not exceed a natural number $n$, then the cardinality of the ultraproduct does not exceed $n$.

Prove that every filtered product of infinite systems is infinite.

Suppose that for every $n$ the family $\left\{ \mathfrak {M}_{i}\right\}_{i \in I}$ contains only finitely many systems of cardinality $n$. Prove that in this case the ultraproduct $\prod_{i \in I} \mathfrak {M}_{i} / D$ with respect to a nonprincipal ultrafilter $D$ is infinite.

Prove that if $\left|M_{i}\right| \leq \left|N_{i}\right|$ for every $i \in I$, then for every filter $D$ over $I$,

Let all the factors $\mathfrak {M}_{i}$ ($i \in \mathbb {N}$) be finite, let $D$ be an ultrafilter over $\mathbb {N}$, and let $\left\{ i \mid \left|M_{i}\right| = n\right\} \notin D$ for every $n \in \mathbb {N}$. Prove that $\prod_{i \in \mathbb {N}} \mathfrak {M}_{i}$ has the cardinality of the continuum.

Let all the factors $\mathfrak {M}_{i}$ ($i \in \mathbb {N}$) be finite or countable, let $D$ be a nonprincipal ultrafilter over the set of natural numbers $\mathbb {N}$, and let $\left\{ i \mid \left|M_{i}\right| = n\right\} \notin D$ for every natural number $n$. Prove that the cardinality of $\prod_{i \in \mathbb {N}} \mathfrak {M}_{i} / D$ equals the continuum.

Prove that if $D$ is not a countably complete ultrafilter over $I$, and for every natural number $n$ we have $\left\{ i \mid \left|M_{i}\right| = n\right\} \notin D$, then $\prod_{i \in I} \mathfrak {M}_{i} / D$ has cardinality at least the continuum.

Let all $\mathfrak {M}_{i}$ ($i \in I$) be countable, and let $D$ be a countably complete ultrafilter over $I$. Prove that $\left|\prod_{i \in I} \mathfrak {M}_{i} / D\right| = \aleph_{0}$.

Let $\langle A_{i}; \leq \rangle$ be partially ordered sets ($i \in \mathbb {N}$), where $A_{i}$ contains $i$ minimal elements. What is the cardinality of the set of minimal elements in the ultraproduct $\prod_{i \in \mathbb {N}} \mathfrak {A}_{i} / D$, if $D$ is a nonprincipal ultrafilter over $\mathbb {N}$?

Prove that for every infinite set $I$ there exists a filter $D$ over $I$ such that for every filter $D_{1}$ over $I$ containing $D$, and every infinite system $\mathfrak {M}$,

Prove that for every infinite algebraic system $\mathfrak {M}$ and every given cardinality $m$, there exists an ultrapower of the system $\mathfrak {M}$ whose cardinality is greater than $m$.

Prove that if $D$ is a not countably complete ultrafilter over $I$ and the algebraic system $\mathfrak {M}$ is infinite, then the natural embedding $\varphi : M \to M^{I} / D$, i.e., $\varphi (a) = f / D$, where $f(i) = a$ for all $i \in I$, is not a one-to-one correspondence between $M$ and $M^{I} / D$.