“Funcoids and Reloids” update
I updated the online draft of the article “Funcoids and Reloids”. The main feature of this update is that I qualified lattice theoretic operations and direct products of filters with indexes indicating that these are used over the sets of funcoids and reloids. That change should make the article easier to read. But (in a […]
Funcoids and Reloids – two conjectures solved
Updated version of Funcoids and Reloids article contains two counter-examples which constitute solutions of two former open problems. It is proved that: There exist atomic reloids whose composition is non-atomic (and not empty). There exists an atomic reloid which is not monovalued.
Funcoids and Reloids updated to match updated filters theory
I updated the online draft of the article “Funcoids and Reloids” to match updated theory of filters (see also this wiki). The main feature of the new version is using filter objects instead of plain filters. Also in the new version of Funcoids and Reloids removed some theorems which are present in the theory of […]
Filters on Posets and Generalizations: Submitted to another math journal
I submitted to “Topology” math journal by email the manuscript Filters on Posets and Generalizations. In the email I asked them to confirm receipt of the email as soon as they receive it. Until now there were no response from “Topology” math journal. So I count them unresponsive and submitted the same work to an […]
Filters on Posets and Generalizations – preprint
I submitted a preprint of the “Filters on Posets and Generalizations” article for peer-review and publication in Topology journal. The current version of this article is located at this URL.
Co-separability of filter objects – solved
I solved a problem earlier formulated in this blog post. A solution (of a slightly more general problem) can be found at this wiki page.
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 […]
Exposition: Complementive filters are complete lattice
(In a past version of this article I erroneously concluded that our main conjecture follows from join-closedness of $latex {Z (D \mathcal{A})}&fg=000000$.) 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 […]