Another conjecture about reloids and funcoids ★★

Author(s): Porton

Definition   $ \square f = \bigcap^{\mathsf{RLD}} \mathrm{up}^{\Gamma (\operatorname{Src} f ; \operatorname{Dst} f)} f $ for reloid $ f $.
Conjecture   $ (\mathsf{RLD})_{\Gamma} f = \square (\mathsf{RLD})_{\mathrm{in}} f $ for every funcoid $ f $.

Note: it is known that $ (\mathsf{RLD})_{\Gamma} f \ne \square (\mathsf{RLD})_{\mathrm{out}} f $ (see below mentioned online article).

Keywords:

One-way functions exist ★★★★

Author(s):

Conjecture   One-way functions exist.

Keywords: one way function

Funcoid corresponding to reloid through lattice Gamma ★★

Author(s): Porton

Conjecture   For every reloid $ f $ and $ \mathcal{X} \in \mathfrak{F} (\operatorname{Src} f) $, $ \mathcal{Y} \in \mathfrak{F} (\operatorname{Dst} f) $:
    \item $ \mathcal{X} \mathrel{[(\mathsf{FCD}) f]} \mathcal{Y}   \Leftrightarrow \forall F \in \operatorname{up}^{\Gamma (\operatorname{Src} f ; \operatorname{Dst}   f)} f : \mathcal{X} \mathrel{[F]} \mathcal{Y} $; \item $ \langle (\mathsf{FCD}) f \rangle \mathcal{X} = \bigcap_{F   \in \operatorname{up}^{\Gamma (\operatorname{Src} f ; \operatorname{Dst} f)} f} \langle F \rangle   \mathcal{X} $.

It's proved by me in this online article.

Keywords: funcoid corresponding to reloid

Restricting a reloid to lattice Gamma before converting it into a funcoid ★★

Author(s): Porton

Conjecture   $ (\mathsf{FCD}) f = \bigcap^{\mathsf{FCD}} (\Gamma (A ; B) \cap \operatorname{GR} f) $ for every reloid $ f \in \mathsf{RLD} (A ; B) $.

Keywords: funcoid corresponding to reloid; funcoids; reloids

Inner reloid through the lattice Gamma ★★

Author(s): Porton

Conjecture   $ (\mathsf{RLD})_{\operatorname{in}} f = \bigcap^{\mathsf{RLD}} \operatorname{up}^{\Gamma (\operatorname{Src} f ; \operatorname{Dst} f)} f $ for every funcoid $ f $.

Counter-example: $ (\mathsf{RLD})_{\operatorname{in}} f \sqsubset \bigcap^{\mathsf{RLD}} \operatorname{up}^{\Gamma (\operatorname{Src} f ; \operatorname{Dst} f)} f $ for the funcoid $ f = (=)|_\mathbb{R} $ is proved in this online article.

Keywords: filters; funcoids; inner reloid; reloids