funcoid


Direct proof of a theorem about compact funcoids ★★

Author(s): Porton

Conjecture   Let $ f $ is a $ T_1 $-separable (the same as $ T_2 $ for symmetric transitive) compact funcoid and $ g $ is a uniform space (reflexive, symmetric, and transitive endoreloid) such that $ ( \mathsf{\tmop{FCD}}) g = f $. Then $ g = \langle f \times f \rangle^{\ast} \Delta $.

The main purpose here is to find a direct proof of this conjecture. It seems that this conjecture can be derived from the well known theorem about existence of exactly one uniformity on a compact set. But that would be what I call an indirect proof, we need a direct proof instead.

The direct proof may be constructed by correcting all errors an omissions in this draft article.

Direct proof could be better because with it we would get a little more general statement like this:

Conjecture   Let $ f $ be a $ T_1 $-separable compact reflexive symmetric funcoid and $ g $ be a reloid such that
    \item $ ( \mathsf{\tmop{FCD}}) g = f $; \item $ g \circ g^{- 1} \sqsubseteq g $.

Then $ g = \langle f \times f \rangle^{\ast} \Delta $.

Keywords: compact space; compact topology; funcoid; reloid; uniform space; uniformity

Distributivity of a lattice of funcoids is not provable without axiom of choice

Author(s): Porton

Conjecture   Distributivity of the lattice $ \mathsf{FCD}(A;B) $ of funcoids (for arbitrary sets $ A $ and $ B $) is not provable in ZF (without axiom of choice).

A similar conjecture:

Conjecture   $ a\setminus^{\ast} b = a\#b $ for arbitrary filters $ a $ and $ b $ on a powerset cannot be proved in ZF (without axiom of choice).

Keywords: axiom of choice; distributive lattice; distributivity; funcoid; reverse math; reverse mathematics; ZF; ZFC

Distributivity of inward reloid over composition of funcoids ★★

Author(s): Porton

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

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

Funcoid corresponding to inward reloid ★★

Author(s): Porton

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

Keywords: funcoid; inward reloid; reloid

Intersection of complete funcoids ★★

Author(s): Porton

Conjecture   If $ f $, $ g $ are complete funcoids (generalized closures) then $ f \cap^{\mathsf{\tmop{FCD}}} g $ is a complete funcoid (generalized closure).

Keywords: complete funcoid; funcoid; generalized closure

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