Upgrading a multifuncoid (Solved)

Importance: Medium ✭✭

Author(s):

Subject:

Keywords:

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:

$\left( \tmop{val} f \right)_i L$

$i \in \tmop{dom} \mathfrak{A}$

$L \in \prod \mathfrak{A}|_{\left( \tmop{dom} \mathfrak{A} \right) \setminus \left\{ i \right\}}$

\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.