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 […]
Are principal filters the center of the lattice of filters?
This conjecture has a seemingly trivial case when $latex \mathcal{A}$ is a principal filter. When I attempted to prove this seemingly trivial case I stumbled over a looking simple but yet unsolved problem: Let $latex U$ is a set. A filter (on $latex U$) $latex \mathcal{F}$ is by definition a non-empty set of subsets of […]
Complete lattice generated by a partitioning – finite meets
I conjectured certain formula for the complete lattice generated by a strong partitioning of an element of complete lattice. Now I have found a beautiful proof of a weaker statement than this conjecture. (Well, my proof works only in the case of distributive lattices, but the case of non-distributive lattices is outside of my research […]
Complete lattice generated by a partitioning of a lattice element
In this post I defined strong partitioning of an element of a complete lattice. For me it was seeming obvious that the complete lattice generated by the set $latex S$ where $latex S$ is a strong partitioning is equal to $latex \left\{ \bigcup{}^{\mathfrak{A}}X | X\in\mathscr{P}S \right\}$. But when I actually tried to write down the […]
Partitioning elements of distributive and finite lattices
I proposed this open problem for the next polymath project. Now I will consider some its special simple cases.
Proposal: Partitioning a lattice element
I’ve given two different definitions for partitioning an element of a complete lattice (generalizing partitioning of a set). I called them weak partitioning and strong partitioning. The problem is whether these two definitions are equivalent for all complete lattices, or if are not then under which additional conditions these are equivalent. (I suspect these may […]
Partitioning of a lattice element: a conjecture
Let $latex \mathfrak{A}$ is a complete lattice. Let $latex a\in\mathfrak{A}$. I will call weak partitioning of $latex a$ a set $latex S\in\mathscr{P}\mathfrak{A}\setminus\{0\}$ such that $latex \bigcup{}^{\mathfrak{A}}S = a \text{ and } \forall x\in S: x\cap^{\mathfrak{A}}\bigcup{}^{\mathfrak{A}}(S\setminus\{x\}) = 0$. I will call strong partitioning of $latex a$ a set $latex S\in\mathscr{P}\mathfrak{A}\setminus\{0\}$ such that $latex \bigcup{}^{\mathfrak{A}}S = a […]
Proposal: Conjecture about complementive filters
Earlier I proposed finishing writing this manuscript as a polymath project. But the manuscript contains (among other) this conjecture which can be reasonably separated into an its own detached polymath project.
Do filters complementive to a given filter form a complete lattice?
Let $latex U$ is a set. A filter (on $latex U$) $latex \mathcal{F}$ is by definition a non-empty set of subsets of $latex U$ such that $latex A,B\in\mathcal{F} \Leftrightarrow A\cap B\in\mathcal{F}$. Note that unlike some other authors I do not require $latex \varnothing\notin\mathcal{F}$. I will denote $latex \mathscr{F}$ the lattice of all filters (on $latex […]
Isomorphic filters – open problems
For filters on sets defined equivalence relation being isomorphic. Posed some open problems like this: are every two nontrivial ultrafilters isomorphic?