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: 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.
Proposal: Filters on Posets and Generalizations
I propose to collaboratively finish writing my manuscript “Filters on Posets and Generalizations” which should become the exhaustive reference text about filters on posets, filters on lattices, and generalizations thereof. I have setup this wiki for this purpose.
Filters on Posets at Google Knol
I removed this Knol. The development of “Filters on Posets and Generalizations” happens on wikidot.com instead. I decided to put my draft article “Filters on Posets and Generalizations” at Google Knol to be edited collaboratively by the Internet math community. The current PDF draft (in fact a very rough draft) is at this URL. Here […]
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?