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.
I take a vacation in my research work
I decided to dedicate my free (of working as a programmer) time to write a book about religion (What book? It will be a surprise.) So in a few nearby months I am going to not continue my math research. I am going to return to math research after I will finish that religious book. […]
Defined limit of discontinuous functions!
In my online draft article “Convergence of funcoids” at my Algebraic General Topology site is now defined limit of arbitrary (not necessarily continuous) functions (under certain conditions). Thus mathematical analysis goes to the next stage, non-continuous analysis. Please nominate me for Abel Prize.
Erroneous theorem became a conjecture
I mistakenly used yet unproved statement that $latex \mathrm{up}\,f$ (taken on the filtrator of funcoids) is a filter for every funcoid $latex f$ in proof of a theorem. So after I found this error I downgrade this theorem to the status of conjecture: Conjecture $latex (\mathsf{FCD}) (\mathsf{RLD})_{\mathrm{out}} f = f$ for every funcoid $latex f$.
Erroneous lemma corrected
In “Funcoids and Reloids” online draft there was an erroneous lemma: Lemma For every two sets $latex S$ and $latex T$ of binary relations and every set $latex A$ $latex \bigcap {\nobreak}^{\mathfrak{F}} S = \bigcap {\nobreak}^{\mathfrak{F}} T \Rightarrow \bigcap {\nobreak}^{\mathfrak{F}} \{ \langle F \rangle A | F \in S \} = \bigcap{\nobreak}^{\mathfrak{F}} \{ \left\langle G […]
Errors in Funcoids and Reloids corrected
I corrected some errors in “Funcoids and Reloids” online draft. The main error was messing $latex \subseteq$ and $latex \supseteq$ in the theorem about continuing a function defined on atomic filter objects till $latex \langle f\rangle$ for a funcoid $latex f$ (currently the Theorem 54 but theorem numbering will be changed in the future). I […]
Resubmit to Documenta Mathematica
I submitted the preprint of my article “Filters on Posets and Generalizations” to Documenta Mathematica math journal but so far received no reply. So I sent submission to an other editor of the same journal.
False proof of an open problem
I earlier proclaimed that I positively solved this conjecture: Conjecture $latex f\cap^{\mathsf{FCD}} g = f\cap g$ for every binary relations $latex f$ and $latex g$. There were error in my proof and I deleted it. See the draft of Funcoids and Reloids article for details on the current state of the problem.