![](/files/happy5.png)
Upgrading a completary multifuncoid
Let be a set,
be the set of filters on
ordered reverse to set-theoretic inclusion,
be the set of principal filters on
, let
be an index set. Consider the filtrator
.
Conjecture If
is a completary multifuncoid of the form
, then
is a completary multifuncoid of the form
.
![$ f $](/files/tex/43374150a8a220f67049937b9790b7d28eb17fb9.png)
![$ \mathfrak{P}^n $](/files/tex/81d0b4bc571faee2e8f352e99db8b54be6f9bb4f.png)
![$ E^{\ast} f $](/files/tex/93b48164cf5af924121c451b6bb17268ae140bae.png)
![$ \mathfrak{F}^n $](/files/tex/34e3ea9e5283eb80b19513453991c07af9c98f8a.png)
See below for definition of all concepts and symbols used to in this conjecture.
Refer to this Web site for the theory which I now attempt to generalize.
Definition A filtrator is a pair
of a poset
and its subset
.
![$ \left( \mathfrak{A}; \mathfrak{Z} \right) $](/files/tex/ec724698dc0160543f8ac2a504055a04020a1057.png)
![$ \mathfrak{A} $](/files/tex/702b76abd81b24daaf0e6bc2a191fb964b09d1b2.png)
![$ \mathfrak{Z} $](/files/tex/1a0a119ad5a02a95e079e5ab5e33e3b1a5052f73.png)
Having fixed a filtrator, we define:
Definition
for every
.
![$ \ensuremath{\operatorname{up}}x = \left\{ Y \in \mathfrak{Z} \hspace{0.5em} | \hspace{0.5em} Y \geqslant x \right\} $](/files/tex/e0dd312a0d02bed5a2531f7c04f9a9c925a7e427.png)
![$ X \in \mathfrak{A} $](/files/tex/9cbd101f45e7f6ced648f9546606ca69e2330433.png)
Definition
(upgrading the set
) for every
.
![$ E^{\ast} K = \left\{ L \in \mathfrak{A} \hspace{0.5em} | \hspace{0.5em} \ensuremath{\operatorname{up}}L \subseteq K \right\} $](/files/tex/5ac0bb072b8567781c862aca1decbe1f100bc971.png)
![$ K $](/files/tex/fe461893b9f87593e8b79d83455b531cd9f29913.png)
![$ K \in \mathscr{P} \mathfrak{Z} $](/files/tex/296ef11d0a13a6388c1efe4b2d778a7cf0aae1ec.png)
Definition Let
is a family of join-semilattice. A completary multifuncoid of the form
is an
such that we have that:
![$ \mathfrak{A} $](/files/tex/702b76abd81b24daaf0e6bc2a191fb964b09d1b2.png)
![$ \mathfrak{A} $](/files/tex/702b76abd81b24daaf0e6bc2a191fb964b09d1b2.png)
![$ f \in \mathscr{P} \prod \mathfrak{A} $](/files/tex/549eb9137ca23d96fcd29b48666a1612a8a5818b.png)
- \item
![$ 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 $](/files/tex/3f6a3cca43a2afc884cf51e4abb225b148a02c79.png)
![$ L_0, L_1 \in \prod \mathfrak{A} $](/files/tex/e2d9708b1dadb9f03d9703d4459f860fdbde1ac1.png)
\item If and
for some
then
.
is a function space over a poset
that is
for
.
For finite this problem is equivalent to Upgrading a multifuncoid .
It is not hard to prove this conjecture for the case using the techniques from this my article. But I failed to prove it for
and above.
Bibliography
* indicates original appearance(s) of problem.