Porton, Victor

Pseudodifference of filter objects ★★

Author(s): Porton

Let $U$ is a set. A filter $\mathcal{F}$ (on $U$ ) is a non-empty set of subsets of $U$ such that $A, B \in \mathcal{F} \Leftrightarrow A \cap B \in \mathcal{F}$ . Note that unlike some other authors I do not require $\emptyset \notin \mathcal{F}$ .

I will call the set of filter objects the set of filters ordered reverse to set theoretic inclusion of filters, with principal filters equated to the corresponding sets. See here for the formal definition of filter objects. I will denote $(\operatorname{up} a)$ the filter corresponding to a filter object $a$ . I will denote the set of filter objects (on $U$ ) as $\mathfrak{F}$ .

I will denote $(\operatorname{atoms} a)$ the set of atomic lattice elements under a given lattice element $a$ . If $a$ is a filter object, then $(\operatorname{atoms} a)$ is essentially the set of ultrafilters over $a$ .

Problem Which of the following expressions are pairwise equal for all $a, b \in \mathfrak{F}$ for each set $U$ ? (If some are not equal, provide counter-examples.)

$\bigcap^{\mathfrak{F}} \left\{ z \in \mathfrak{F} | a \subseteq b \cup^{\mathfrak{F}} z \right\}$

\item $\bigcup^{\mathfrak{F}} \left\{ z \in \mathfrak{F} | z \subseteq a \wedge z \cap^{\mathfrak{F}} b = \emptyset \right\}$ ;

\item $\bigcup^{\mathfrak{F}} (\operatorname{atoms} a \setminus \operatorname{atoms} b)$ ;

\item $\bigcup^{\mathfrak{F}} \left\{ a \cap^{\mathfrak{F}} (U\setminus B) | B \in \operatorname{up} b \right\}$ .

Keywords: filters; pseudodifference

Posted by porton
updated May 19th, 2012

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Unsorted

Do filters complementive to a given filter form a complete lattice? ★★

Author(s): Porton

Let $U$ is a set. A filter (on $U$ ) $\mathcal{F}$ is by definition a non-empty set of subsets of $U$ such that $A,B\in\mathcal{F} \Leftrightarrow A\cap B\in\mathcal{F}$ . Note that unlike some other authors I do not require $\varnothing\notin\mathcal{F}$ . I will denote $\mathscr{F}$ the lattice of all filters (on $U$ ) ordered by set inclusion.

Let $\mathcal{A}\in\mathscr{F}$ is some (fixed) filter. Let $D=\{\mathcal{X}\in\mathscr{F} | \mathcal{X}\supseteq \mathcal{A}\}$ . Obviously $D$ is a bounded lattice.

I will call complementive such filters $\mathcal{C}$ that:

$\mathcal{C}\in D$ ;
$\mathcal{C}$ is a complemented element of the lattice $D$ .

Conjecture The set of complementive filters ordered by inclusion is a complete lattice.

Keywords: complete lattice; filter

Posted by porton
updated July 31st, 2009

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Topology

Monovalued reloid restricted to atomic filter ★★

Author(s): Porton

Conjecture A monovalued reloid restricted to an atomic filter is atomic or empty.

Weaker conjecture:

Conjecture A (monovalued) function restricted to an atomic filter is atomic or empty.

Keywords: monovalued reloid

Posted by porton
updated February 5th, 2009

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Topology

Atomic reloids are monovalued ★★

Author(s): Porton

Conjecture Atomic reloids are monovalued.

Keywords: atomic reloid; monovalued reloid; reloid

Posted by porton
updated February 4th, 2009

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Topology

Composition of atomic reloids ★★

Author(s): Porton

Conjecture Composition of two atomic reloids is atomic or empty.

Keywords: atomic reloid; reloid

Posted by porton
updated February 4th, 2009

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Topology

S(S(f)) = S(f) for reloids ★★

Author(s): Porton

Question $S(S(f)) = S(f)$ for every endo-reloid $f$ ?

Keywords: reloid

Posted by porton
updated May 8th, 2008

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Topology

Reloid corresponding to funcoid is between outward and inward reloid ★★

Author(s): Porton

Conjecture For any funcoid $f$ and reloid $g$ having the same source and destination $( \mathsf{\tmop{RLD}})_{\tmop{out}} f \subseteq g \subseteq ( \mathsf{\tmop{RLD}})_{\tmop{in}} f \Leftrightarrow ( \mathsf{\tmop{FCD}}) g = f.$

Keywords: funcoid; inward reloid; outward reloid; reloid

Posted by porton
updated September 2nd, 2007

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Topology

Distributivity of union of funcoids corresponding to reloids ★★

Author(s): Porton

Conjecture $\bigcup \left\langle ( \mathsf{\tmop{FCD}}) \right\rangle S = ( \mathsf{\tmop{FCD}}) \bigcup S$ if $S\in\mathscr{P}\mathsf{RLD}(A;B)$ is a set of reloids from a set $A$ to a set $B$ .