Connectors and generalized connectedness – draft
I updated online draft of the article “Connectors and generalized connectedness” which considers a generalization of topological connectedness, graph connectedness, proximal connectedness, uniform connectedness, etc.
Draft about connectedness
I put online a preliminary draft of my article Connectors and generalized connectedness. This preliminary draft is yet unreadable. The primary reason why I put this unreadable preliminary draft online is that I want to refer to it from this MathOverflow question. From the article: This article is an unreadable preliminary draft. Some definitions and […]
A distributive law for filter objects
I recently proved the following conjecture (now a theorem): Theorem $latex A\cap^{\mathfrak{F}}\bigcup{}^{\mathfrak{F}}S = \bigcup{}^{\mathfrak{F}} \{ A\cap^{\mathfrak{F}} X | X\in S \}$ for every set $latex A\in\mathscr{P}\mho$ and every $latex S\in\mathscr{P}\mathfrak{F}$ where $latex \mathfrak{F}$ is the set of filter objects on some set $latex \mho$. This theorem is a direct consequence of the following lemma: Lemma […]
My research plans shifted
Yesterday I wrote that I next thing which I will research are n-ary funcoids and n-ary reloids. It seems that (n+m)-ary funcoid can be split into a funcoid acting from n-ary funcoids to m-ary funcoids (similarly to (n+m)-ary relation can be split into a binary relation between n-ary tuples and m-ary tuples). But this funcoid […]
My research plan
I ‘ve said that I take a vacation in my math research work in order to write a religious book. Unexpectedly quickly I have already finished to write and publish this book and return to my mathematical research. Now having researched enough about funcoids and reloids (despite of there are yet several open problems in […]
Funcoids and reloids, a Galois connection
I proved that $latex (\mathsf{FCD})$ is the lower adjoint of $latex (\mathsf{RLD})_{\mathrm{in}}$. Also from this follows that $latex (\mathsf{FCD})$ preserves all suprema and $latex (\mathsf{RLD})_{\mathrm{in}}$ preserves all infima. See Algebraic General Topology and specifically Funcoids and Reloids online article.
A counterexample: Funcoid corresponding to outer reloid
I found a counter-example to the following conjecture. Conjecture $latex (\mathsf{FCD}) (\mathsf{RLD})_{\mathrm{out}} f = f$ for every funcoid $latex f$. The counterexample is $latex f = {(=)}|_{\Omega}$ where $latex \Omega$ is the Fréchet filter. See Algebraic General Topology and in particular Funcoids and Reloids online article, the section Some counter-examples for this counterexample.
A counterexample against “Meet of discrete funcoids is discrete”
I found a counterexample to the following conjecture: Conjecture $latex f\cap^{\mathsf{FCD}} g = f\cap g$ for every binary relations $latex f$ and $latex g$. The counter-example is $latex f = {(=)}|_{\mho}$ and $latex g = \mho\times\mho \setminus f$. I proved $latex f \cap^{\mathsf{FCD}} g = {(=)} |_{\Omega}$ (where $latex \Omega$ is the Frechet filter object). […]
A theorem generalized
I generalized a theorem in the preprint article “Filters on posets and generalizations” on my Algebraic General Topology site. The new theorem is formulated as following: Theorem If $latex (\mathfrak{A}; \mathfrak{Z})$ is a join-closed filtrator and $latex \mathfrak{A}$ is a meet-semilattice and $latex \mathfrak{Z}$ is a complete lattice, then $latex \mathrm{Cor}’ (a \cap^{\mathfrak{A}} b) = […]
Filters article sent to Moscow Mathematical Journal
My second submit to Documenta Mathematica journal of “Filters on Posets and Generalizations” preprint was unanswered in reasonable amount of time. As such I submitted it to an other journal, Moscow Mathematical Journal.