Filter rebase generalized
I have re-defined filter rebase. Now it is defined for arbitrary filter $latex \mathcal{A}$ on some set $latex \mathrm{Base}(\mathcal{A})$ and arbitrary set $latex A$. The new definition is: $latex \mathcal{A}\div A = \{ X\in\mathscr{P}A \mid \exists Y\in\mathcal{A}: Y\cap A\subseteq X \}$. It is shown that for the special case of $latex \forall X\in\mathcal{A}:X\subseteq A$ the […]
A new easy theorem
I added a new easy to prove proposition to my book: Proposition An endofuncoid $latex f$ is $latex T_{1}$-separable iff $latex \mathrm{Cor}\langle f\rangle^{\ast}\{x\}\sqsubseteq\{x\}$ for every $latex x\in\mathrm{Ob}\, f$.
A math question
What are necessary and sufficient conditions for $latex \mathrm{up}\, f$ to be a filter for a funcoid $latex f$?
New easy theorem and its consequence
I’ve added to my book a new easy to prove theorem and its corollary: Theorem If $latex f$ is a (co-)complete funcoid then $latex \mathrm{up}\, f$ is a filter. Corollary If $latex f$ is a (co-)complete funcoid then $latex \mathrm{up}\, f = \mathrm{up} (\mathsf{RLD})_{\mathrm{out}} f$. If $latex f$ is a (co-)complete reloid then $latex \mathrm{up}\, […]
About “Each regular paratopological group is completely regular” article
In this blog post I consider my attempt to rewrite the article “Each regular paratopological group is completely regular” by Taras Banakh, Alex Ravsky in a more abstract way using my theory of reloids and funcoids. The following is a general comment about reloids and funcoids as defined in my book. If you don’t understand […]
An important conjecture about funcoids. Version 2
This conjecture appeared to be false. Now I propose an alternative conjecture: Let $latex A$, $latex B$ be sets. Conjecture Funcoids $latex f$ from $latex A$ to $latex B$ bijectively corresponds to the sets $latex R$ of pairs $latex (\mathcal{X}; \mathcal{Y})$ of filters (on $latex A$ and $latex B$ correspondingly) that $latex R$ is nonempty. […]
An important conjecture about funcoids
Just a few minutes ago I’ve formulated a new important conjecture about funcoids: Let $latex A$, $latex B$ be sets. Conjecture Funcoids $latex f$ from $latex A$ to $latex B$ bijectively corresponds to the sets $latex R$ of pairs $latex (\mathcal{X}; \mathcal{Y})$ of filters (on $latex A$ and $latex B$ correspondingly) that $latex R$ is […]
Join of transitive reloids (a conjecture in uniformity theory)
Conjecture Join of a set $latex S$ on the lattice of transitive reloids is the join (on the lattice of reloids) of all compositions of finite sequences of elements of $latex S$. It was expired by theorem 2.2 in “Hans Weber. On lattices of uniformities”. There is a similar conjecture for funcoids (instead of reloids). […]
Galois connections are related with pointfree funcoids!
I call pointfree funcoids (see my free e-book) between boolean lattices as boolean funcoids. I have proved that: Theorem Let $latex \mathfrak{A}$ and $latex \mathfrak{B}$ be complete boolean lattices. Then $latex \alpha$ is the first component of a boolean funcoid iff it is a lower adjoint (in the sense of Galois connections between posets). Does […]
I’ve partially proved a conjecture
The following is a conjecture: Conjecture The set of pointfree funcoids between two boolean lattices is itself a boolean lattice. Today I have proved its special case: Theorem The set of pointfree funcoids between a complete boolean lattice and an atomistic boolean lattice is itself a boolean lattice. It is a very weird theorem because […]