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Porton, Victor


Topology

Upgrading a multifuncoid ★★

Author(s): Porton

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.

Keywords:

Posted by porton
updated October 9th, 2011
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Topology

Strict inequalities for products of filters ★

Author(s): Porton

Conjecture   $ \mathcal{A} \times^{\mathsf{\ensuremath{\operatorname{RLD}}}}_F \mathcal{B} \subset \mathcal{A} \ltimes \mathcal{B} \subset \mathcal{A} \times^{\mathsf{\ensuremath{\operatorname{RLD}}}} \mathcal{B} $ for some filter objects $ \mathcal{A} $, $ \mathcal{B} $. Particularly, is this formula true for $ \mathcal{A} = \mathcal{B} = \Delta \cap \uparrow^{\mathbb{R}} \left( 0 ; + \infty \right) $?

A weaker conjecture:

Conjecture   $ \mathcal{A} \times^{\mathsf{\ensuremath{\operatorname{RLD}}}}_F \mathcal{B} \subset \mathcal{A} \ltimes \mathcal{B} $ for some filter objects $ \mathcal{A} $, $ \mathcal{B} $.

Keywords: filter products

Posted by porton
updated August 9th, 2011
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Topology

Join of oblique products ★★

Author(s): Porton

Conjecture   $ \left( \mathcal{A} \ltimes \mathcal{B} \right) \cup \left( \mathcal{A} \rtimes \mathcal{B} \right) = \mathcal{A} \times^{\mathsf{\ensuremath{\operatorname{RLD}}}} \mathcal{B} $ for every filter objects $ \mathcal{A} $, $ \mathcal{B} $.

Keywords: filter; oblique product; reloidal product

Posted by porton
updated August 9th, 2011
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Topology

Outer reloid of restricted funcoid ★★

Author(s): Porton

Question   $ ( \mathsf{RLD})_{\mathrm{out}} (f \cap^{\mathsf{FCD}} ( \mathcal{A} \times^{\mathsf{FCD}} \mathcal{B})) = (( \mathsf{RLD})_{\mathrm{out}} f) \cap^{\mathsf{RLD}} ( \mathcal{A} \times^{\mathsf{RLD}} \mathcal{B}) $ for every filter objects $ \mathcal{A} $ and $ \mathcal{B} $ and a funcoid $ f\in\mathsf{FCD}(\mathrm{Src}\,f; \mathrm{Dst}\,f) $?

Keywords: direct product of filters; outer reloid

Posted by porton
updated December 3rd, 2010
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Topology

Domain and image of inner reloid ★★

Author(s): Porton

Conjecture   $ \ensuremath{\operatorname{dom}}( \mathsf{\ensuremath{\operatorname{RLD}}})_{\ensuremath{\operatorname{in}}} f =\ensuremath{\operatorname{dom}}f $ and $ \ensuremath{\operatorname{im}}( \mathsf{\ensuremath{\operatorname{RLD}}})_{\ensuremath{\operatorname{in}}} f =\ensuremath{\operatorname{im}}f $ for every funcoid $ f $.

Keywords: domain; funcoids; image; reloids

Posted by porton
updated December 2nd, 2010
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Topology

Characterization of monovalued reloids with atomic domains ★★

Author(s): Porton

Conjecture   Every monovalued reloid with atomic domain is either
  1. an injective reloid;
  2. a restriction of a constant function

(or both).

Keywords: injective reloid; monovalued reloid

Posted by porton
updated October 26th, 2010
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Topology

Composition of reloids expressed through atomic reloids ★★

Author(s): Porton

Conjecture   If $ f $ and $ g $ are composable reloids, then $$g \circ f = \bigcup \{G \circ F | F \in \mathrm{atoms}\, f, G \in \mathrm{atoms}\, g \}.$$

Keywords: atomic reloids

Posted by porton
updated October 26th, 2010
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Topology

Outer reloid of direct product of filters ★★

Author(s): Porton

Question   $ ( \mathsf{\tmop{RLD}})_{\tmop{out}} ( \mathcal{A} \times^{\mathsf{\tmop{FCD}}} \mathcal{B}) = \mathcal{A} \times^{\mathsf{\tmop{RLD}}} \mathcal{B} $ for every f.o. $ \mathcal{A} $, $ \mathcal{B} $?

Keywords: direct product of filters; outer reloid

Posted by porton
updated September 4th, 2010
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Unsorted

Chain-meet-closed sets ★★

Author(s): Porton

Let $ \mathfrak{A} $ is a complete lattice. I will call a filter base a nonempty subset $ T $ of $ \mathfrak{A} $ such that $ \forall a,b\in T\exists c\in T: (c\le a\wedge c\le b) $.

Definition   A subset $ S $ of a complete lattice $ \mathfrak{A} $ is chain-meet-closed iff for every non-empty chain $ T\in\mathscr{P}S $ we have $ \bigcap T\in S $.
Conjecture   A subset $ S $ of a complete lattice $ \mathfrak{A} $ is chain-meet-closed iff for every filter base $ T\in\mathscr{P}S $ we have $ \bigcap T\in S $.

Keywords: chain; complete lattice; filter bases; filters; linear order; total order

Posted by porton
updated December 12th, 2009
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Unsorted

Co-separability of filter objects ★★

Author(s): Porton

Conjecture   Let $ a $ and $ b $ are filters on a set $ U $ and $ a\cap b = \{U\} $. Then $$\exists A\in a,B\in b: (\forall X\in a: A\subseteq X \wedge \forall Y\in b: B\subseteq Y \wedge A \cup B = U).$$

See here for some equivalent reformulations of this problem.

This problem (in fact, a little more general version of a problem equivalent to this problem) was solved by the problem author. See here for the solution.

Maybe this problem should be moved to "second-tier" because its solution is simple.

Keywords: filters

Posted by porton
updated September 10th, 2010
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