Importance: Medium ✭✭
Author(s): Porton, Victor
Subject: Topology
Recomm. for undergrads: no
Posted by: porton
on: October 9th, 2011
Solved by: Porton, Victor

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) $.

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

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

Refer to this Web site for the theory which I now attempt to generalize.

I found a really trivial proof of this conjecture. See this my draft article.

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

Having fixed a filtrator, we define:

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} $.
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} $.
Definition   A free star on a join-semilattice $ \mathfrak{A} $ with least element 0 is a set $ S $ such that $ 0 \not\in S $ and \[ \forall A, B \in \mathfrak{A}: \left( A \cup B \in S \Leftrightarrow A      \in S \vee B \in S \right) . \]
Definition   Let $ \mathfrak{A} $ be a family of posets, $ f \in \mathscr{P} \prod   \mathfrak{A} $ ($ \prod \mathfrak{A} $ has the order of function space of posets), $ i \in \ensuremath{\operatorname{dom}}\mathfrak{A} $, $ L \in \prod   \mathfrak{A}|_{\left( \ensuremath{\operatorname{dom}}\mathfrak{A} \right)   \setminus \left\{ i \right\}} $. Then \[ \left( \ensuremath{\operatorname{val}}f \right)_i L = \left\{ X \in      \mathfrak{A}_i \hspace{0.5em} | \hspace{0.5em} L \cup \left\{ (i ; X)      \right\} \in f \right\} . \]
Definition   Let $ \mathfrak{A} $ is a family of posets. A multidimensional funcoid (or multifuncoid for short) of the form $ \mathfrak{A} $ is an $ f \in \mathscr{P} \prod \mathfrak{A} $ such that we have that:
    \item $ \left( \tmop{val} f \right)_i L $ is a free star for every $ i \in     \tmop{dom} \mathfrak{A} $, $ L \in \prod \mathfrak{A}|_{\left( \tmop{dom}     \mathfrak{A} \right) \setminus \left\{ i \right\}} $.

    \item $ f $ is an upper set.

$ \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 $.

It is not hard to prove this conjecture for the case $ \ensuremath{\operatorname{card}}n \leqslant 2 $ using the techniques from this my article. But I failed to prove it for $ \ensuremath{\operatorname{card}}n = 3 $ and above.

* indicates original appearance(s) of problem.


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