Porton, Victor

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:

  1. $ \mathcal{C}\in D $;
  2. $ \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

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

Atomic reloids are monovalued ★★

Author(s): Porton

Conjecture   Atomic reloids are monovalued.

Keywords: atomic reloid; monovalued reloid; reloid

Composition of atomic reloids ★★

Author(s): Porton

Conjecture   Composition of two atomic reloids is atomic or empty.

Keywords: atomic reloid; reloid

S(S(f)) = S(f) for reloids ★★

Author(s): Porton

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

Keywords: reloid

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

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 $.

Keywords: funcoid; infinite distributivity; reloid

Inward reloid corresponding to a funcoid corresponding to convex reloid ★★

Author(s): Porton

Conjecture   $ ( \mathsf{\tmop{RLD}})_{\tmop{in}} ( \mathsf{\tmop{FCD}}) f = f $ for any convex reloid $ f $.

Keywords: convex reloid; funcoid; functor; inward reloid; reloid

Distributivity of outward reloid over composition of funcoids ★★

Author(s): Porton

Conjecture   $ ( \mathsf{\tmop{RLD}})_{\tmop{out}} (g \circ f) = ( \mathsf{\tmop{RLD}})_{\tmop{out}} g \circ ( \mathsf{\tmop{RLD}})_{\tmop{out}} f $ for any composable funcoids $ f $ and $ g $.

Keywords: distributive; distributivity; funcoid; functor; outward reloid; reloid

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