New conjectures about complete funcoids and reloids

After removing an erroneous theorem I posed two new open problems to take its place: Conjecture If $latex f$ is a complete funcoid and $latex R$ is a set of funcoids then $latex f \circ \bigcup {\nobreak}^{\mathsf{FCD}} R = \bigcup {\nobreak}^{\mathsf{FCD}} \langle f \circ \rangle R$. Conjecture If $latex f$ is a complete reloid and […]

“Funcoids and Reloids” contains “Connectedness”

Now Funcoids and Reloids online article contains the section “Connectedness regarding funcoids and reloids” which previously was in a separate article. In this section there are among definitions and theorems a few open problems.

On the definition of compact funcoids

[I found that my computations below are erroneous, namely $latex \mathrm{Cor} \langle f^{-1}\rangle \mathcal{F} \neq \langle \mathrm{CoCompl} f^{-1}\rangle \mathcal{F}$ in general (the equality holds when $latex \mathcal{F}$ is a set).]

An erroneos theorem (now a conjecture)

In my Algebraic General Topology series was a flaw in the proof of the following theorem. So I re-labeled it as a conjecture. Conjecture A filter $latex \mathcal{A}$ is connected regarding a reloid $latex f$ iff it is connected regarding the funcoid $latex (\mathrm{FCD})f$. Other theorems in my manuscripts are not affected by this error.

Funcoids and Reloids updated – complete reloids

I updated online article “Funcoids and Reloids”. The main feature of this update is new section about complete reloids and completion of reloids (with a bunch of new open problems). Also added some new theorems in the section “Completion of funcoids”.

New conjecture about core parts of filters

Conjecture $latex \mathrm{Cor}\bigcup^{\mathfrak{F}} S=\bigcup\langle \mathrm{Cor}\rangle S$ for any set $latex S$ of filter objects on a set. See this wiki site for definitions of used terms.

Chain-meet-closed sets on complete lattices

Let $latex \mathfrak{A}$ is a complete lattice. I will call a filter base a nonempty subset $latex T$ of $latex \mathfrak{A}$ such that $latex \forall a,b\in T\exists c\in T: (c\le a\wedge c\le b)$. I will call a chain (on $latex \mathfrak{A}$) a linearly ordered subset of $latex \mathfrak{A}$. Now as a part my research of […]

Open problem: co-separability of filter objects

Conjecture Let $latex a$ and $latex b$ are filters on a set $latex U$. Then $latex a\cap b = \{U\} \Rightarrow \\ \exists A,B\in\mathcal{P}U: (\forall X\in a: A\subseteq X \wedge \forall Y\in b: B\subseteq Y \wedge A \cup B = U).$ [corrected] This conjecture can be equivalently reformulated in terms of filter objects: Conjecture Let […]

Open problem: Pseudodifference of filters

Let $latex {U}&fg=000000$ is a set. A filter $latex {\mathcal{F}}&fg=000000$ (on $latex {U}&fg=000000$) is a non-empty set of subsets of $latex {U}&fg=000000$ such that $latex {A, B \in \mathcal{F} \Leftrightarrow A \cap B \in \mathcal{F}}&fg=000000$. Note that unlike some other authors I do not require $latex {\emptyset \notin \mathcal{F}}&fg=000000$. I will call the set of […]