Upgrading a completary multifuncoid

 Importance: Medium ✭✭
 Author(s): Porton, Victor
 Subject: Topology
 Keywords:
 Recomm. for undergrads: no
 Posted by: porton on: February 4th, 2012

Let $\mho$ be a set, $\mathfrak{F}$ be the set of filters on $\mho$ ordered reverse to set-theoretic inclusion, $\mathfrak{P}$ be the set of principal filters on $\mho$, let $n$ be an index set. Consider the filtrator $\left( \mathfrak{F}^n ; \mathfrak{P}^n \right)$.

\begin{conjecture} If $f$ is a completary multifuncoid of the form $\mathfrak{P}^n$, then $E^{\ast} f$ is a completary multifuncoid of the form $\mathfrak{F}^n$. \end{conjecture}

See below for definition of all concepts and symbols used to in this conjecture.

Refer to \href[this Web site]{http://www.mathematics21.org/algebraic-general-topology.html} for the theory which I now attempt to generalize.

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\begin{definition} A filtrator is a pair $\left( \mathfrak{A}; \mathfrak{Z} \right)$ of a poset $\mathfrak{A}$ and its subset $\mathfrak{Z}$. \end{definition}

Having fixed a filtrator, we define:

\begin{definition} $\ensuremath{\operatorname{up}}x = \left\{ Y \in \mathfrak{Z} \hspace{0.5em} | \hspace{0.5em} Y \geqslant x \right\}$ for every $X \in \mathfrak{A}$. \end{definition}

\begin{definition} $E^{\ast} K = \left\{ L \in \mathfrak{A} \hspace{0.5em} | \hspace{0.5em} \ensuremath{\operatorname{up}}L \subseteq K \right\}$ (upgrading the set $K$) for every $K \in \mathscr{P} \mathfrak{Z}$. \end{definition}

\begin{definition} Let $\mathfrak{A}$ is a family of join-semilattice. A completary multifuncoid of the form $\mathfrak{A}$ is an $f \in \mathscr{P} \prod \mathfrak{A}$ such that we have that: \begin{enumerate} \item $L_0 \cup L_1 \in f \Leftrightarrow \exists c \in \left\{ 0, 1 \right\}^n : \left( \lambda i \in n : L_{c \left( i_{} \right)} i \right) \in f$ for every $L_0, L_1 \in \prod \mathfrak{A}$.

\item If $L \in \prod \mathfrak{A}$ and $L_i = 0^{\mathfrak{A}_i}$ for some $i$ then $\neg f L$. \end{enumerate} \end{definition}

$\mathfrak{A}^n$ is a function space over a poset $\mathfrak{A}$ that is $a\le b\Leftrightarrow \forall i\in n:a_i\le b_i$ for $a,b\in\mathfrak{A}^n$.

For finite $n$ this problem is equivalent to \OPrefnum[Upgrading a multifuncoid ]{37348}.

It is not hard to prove this conjecture for the case $\ensuremath{\operatorname{card}}n \leqslant 2$ using the techniques from \href[this my article]{http://www.mathematics21.org/binaries/funcoids-reloids.pdf}. But I failed to prove it for $\ensuremath{\operatorname{card}}n = 3$ and above.

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