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…

read moreIn 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\{…

read moreI proposed this open problem for the next polymath project. Now I will consider some its special simple cases.

read moreI’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…

read moreLet $latex \mathfrak{A}$ 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…

read moreIn the framework of ZF formally considered generalizations, such as whole numbers generalizing natural number, rational numbers generalizing whole numbers, real numbers generalizing rational numbers, complex numbers generalizing real numbers, etc. The formal consideration of this may be especially useful for computer…

read moreEarlier 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.

read moreI 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.

read moreI 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…

read moreLet $latex U$ 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…

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