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chap-prod-new.tex
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\chapter{Product of funcoids over a filter}
The following definition is inspired by the usual definition of Tychonoff
product of topological spaces.
\begin{defn}
Let $f$ be an indexed family of funcoids. Let $\mathcal{F}$ be a filter on
$\dom f$.
\[ a \mathrel{\left[ \prod^{[\mathcal{F}]} f \right]} b \Leftrightarrow
\exists N \in \mathcal{F} \forall i \in N : \Pr^{\mathsf{RLD}}_i a
\suprel{f_i} \Pr^{\mathsf{RLD}}_i b \]
for atomic reloids $a$ and $b$.
\end{defn}
\begin{rem}
We are especially interested in the special case when $\mathcal{F}$ is the
cofinite filter. In this case $a \mathrel{\left[ \prod^{[\mathcal{F}]} f
\right]} b$ is defined by the condition that $\Pr^{\mathsf{RLD}}_i a
\suprel{f_i} \Pr^{\mathsf{RLD}}_i $ for an infinite number of
indexes $i$.
\end{rem}
\begin{obvious}
$a \mathrel{\left[ \prod^{\suprel{ \top^{\mathscr{F} (\dom f)} }} f
\right]} b \Leftrightarrow a \mathrel{\left[ \prod^{(A)} f \right]} b$.
\end{obvious}
\begin{prop}
$\neg \left( \mathcal{X} \suprel{f} \mathcal{Y} \right)$ implies $\neg
\left( X \suprel{f} Y \right)$ for some $X \in \up \mathcal{X}$, $Y
\in \up \mathcal{Y}$.
\end{prop}
\begin{proof}
Suppose $\neg \left( \mathcal{X} \suprel{f} \mathcal{Y} \right)$. Then
$\mathcal{Y} \asymp \supfun{f} \mathcal{X}$. Thus by separability of
core for filters $Y \asymp \supfun{f} \mathcal{X}$ for some $Y \in
\up \mathcal{Y}$, that is $\neg \left( \mathcal{X} \suprel{f} Y
\right)$. Apply this result twice.
\end{proof}
\begin{lem}
~
\[ \forall X \in \prod_{i \in D} \up a_i, Y \in \prod_{i \in D}
\up b_i \exists x \in \prod_{i \in D} \mathrm{atoms} \uparrow X_i,
y \in \prod_{i \in D} \mathrm{atoms} \uparrow Y_i \exists N \in
\mathcal{F} \forall j \in N : x_j \mathrel{[f_j]} y_j \]
implies $\exists N \in \mathcal{F} \forall i \in N : a_i \suprel{f_i}
b_i$.
\end{lem}
\begin{proof}
Suppose for the contrary $\neg \left( a_i \suprel{f_i} b_i \right)$ for
all $i \in N$ where $N \in \mathcal{F}$ (i.e. for an infinite number of
indexes if $\mathcal{F}$ is the cofinite filter). Then (lemma above) there
are $X_i \in \up a_i$ and $Y_i \in \up b_i$ such that $\neg
\left( X_i \mathrel{[f_j]^{\ast}} Y_i \right)$ for $i \in N$. Thus $\neg
\left( x_i \suprel{f_i} y_i \right)$ for $i \in N$, contrary to the
condition.
\end{proof}
\begin{prop}
The funcoid $\prod^{[\mathcal{F}]} f$ exists.
\end{prop}
\begin{proof}
We need to prove that
\[ \forall X \in \up a, Y \in \up b \exists x \in \atoms
\uparrow^{\mathsf{RLD}} X, y \in \atoms
\uparrow^{\mathsf{RLD}} Y : x \mathrel{\left[ \prod^{(A 2)} f
\right]} y \]
implies $a \mathrel{\left[ \prod^{[\mathcal{F}]} f \right]} b$.
Equivalently transforming it: \fxwarning{More detailed proof.}
$\forall X \in \up a, Y \in \up b \exists x \in \atoms
\uparrow^{\mathsf{RLD}} X, y \in \atoms
\uparrow^{\mathsf{RLD}} Y \\
\exists N \in \mathcal{F} \forall i \in N : \Pr^{\mathsf{RLD}}_i x
\suprel{f_i} \Pr^{\mathsf{RLD}}_i y$; \\
$\forall X \in \up a, Y \in \up b \exists x \in \prod_{i \in
\dom f} \atoms \uparrow^{\mathsf{RLD}} X_i, y \in
\prod_{i \in \dom f} \atoms \uparrow^{\mathsf{RLD}} Y_i \\
\exists N \in \mathcal{F} \forall i \in N : x_i \suprel{f_i} y_i$;
\[ \forall X \in \prod_{i \in D} \up a_i, Y \in \prod_{i \in D}
\up b_i \exists x \in \prod_{i \in D} \mathrm{atoms} \uparrow X_i,
y \in \prod_{i \in D} \mathrm{atoms} \uparrow Y_i \exists N \in
\mathcal{F} \forall j \in N : x_j \mathrel{[f_j]} y_j \]
where $D = \dom f$.
Thus by the lemma $\exists N \in \mathcal{F} \forall i \in N : a_i
\suprel{f_i} b_i$, that is $a \suprel{\prod^{[\mathcal{F}]} f} b$.
\end{proof}
\fxnote{TODO: when $\Pr_j \prod^{[\mathcal{F}]}_{i\in D} a_i = a_j$?}
\section{More on product of reloids}
\fxwarning{Move this to a more appropriate place.}
\begin{defn}
$\prod^{(Y)}_{i \in \dom f} f = \prod^{(A)}_{i \in \dom f}
\tofcd f$ for an indexed family~$f$ of reloids.
\end{defn}
\begin{prop}
\[ a \mathrel{\left[ \prod^{(Y)}_{i \in \dom f} f \right]} b
\Leftrightarrow \forall i \in \dom f : f_i \nasymp
\Pr^{\mathsf{RLD}}_i a \times^{\mathsf{RLD}}
\Pr^{\mathsf{RLD}}_i b. \]
\end{prop}
\begin{proof}
$f_i \nasymp \Pr^{\mathsf{RLD}}_i a \times^{\mathsf{FCD}} \Pr^{\mathsf{RLD}}_i b
\Leftrightarrow \tofcd f_i \sqsupseteq
\Pr^{\mathsf{RLD}}_i a \times^{\mathsf{FCD}} \Pr^{\mathsf{RLD}}_i b \Leftrightarrow
a \mathrel{[\tofcd f_i]} b$.
\end{proof}
\begin{example}
The funcoid $p$ described by the formula (for atomic reloids $a$ and $b$)
\[ a \mathrel{\mathrel{p}} b \Leftrightarrow \forall i \in \dom f : f_i
\sqsupseteq \Pr^{\mathsf{RLD}}_i a \times^{\mathsf{RLD}}
\Pr^{\mathsf{RLD}}_i b \]
does not exist (in general), even if we restrict to $2$-indexed families
only.
\end{example}
\begin{proof}
For the case if $f = \llbracket v , w \rrbracket$ is a $2$-indexed family of
reloids, the formula which we need to disprove takes the form:
\[ a \mathrel{\mathrel{p}} b \Leftrightarrow v \sqsupseteq \dom a
\times^{\mathsf{RLD}} \dom b \wedge w \sqsupseteq \im
a \times^{\mathsf{RLD}} \im b. \]
Take $v = w = 1^{\mathbf{Rel}}$ on an infinite set. Suppose for the
contrary $p$ exists and is a funcoid. Then
\[ \forall X \in \up a, Y \in \up b \exists x \in \atoms
\uparrow X, y \in \atoms \uparrow Y : x \mathrel{\mathrel{p}} y \Rightarrow
a \mathrel{\mathrel{p}} b. \]
For a counter-example take $a = b$ to be a nontrivial ultrafilter. Then for
every $X \in \up a$, $Y \in \up b$ take $x = y$ to be singletons
on $X \cap Y$. We have $x \mathrel{\mathrel{p}} y$, but not $a \mathrel{\mathrel{p}} b$.
\end{proof}