My math sickness
I’m sick with the following: I repeatedly formulate conjectures which have trivial counterexamples and am stuck attempting to prove these true not seeing counterexamples. I should less rely on my ideas what is a true conjecture. I just need to become humbler and less proud. Hopefully now I have enough counter-examples for me to follow […]
Counter-example against “inward reloid of corresponding funcoid of a convex reloid” conjecture
I found (a rather trivial) counter-example to one conjecture I considered in the past: Example $latex (\mathsf{RLD})_{\mathrm{in}} (\mathsf{FCD}) f \neq f$ for some convex reloid $latex f$. Proof Let $latex f = (=)$. Then $latex (\mathsf{FCD}) f = (=)$. Let $latex a$ is some nontrivial atomic filter object. Then $latex (\mathsf{RLD})_{\mathrm{in}} (\mathsf{FCD}) (=) \supseteq a […]
Questions about orderings of filters and ultrafilters
I asked on MathOverflow several questions about ordering of filters and ultrafilters. Your participation in this research is welcome.
“Filters on Posets and Generalizations” – updated
Filters on Posets and Generalizations online article updated as an accomplishment of this plan. This is important primarily to extend the category of pointfree funcoids with objects being arbitrary posets (even without least element). That way this category would become more “complete”. To extend that is required a definition of intersecting elements of a poset […]
Intersecting elements of posets without least element
From the preprint of my article “Filters on Posets and Generalizations” (with little rewording): Definition 1. Let $latex \mathfrak{A}$ is a poset with least element $latex 0$. I will call elements $latex a$, $latex b$ in $latex \mathfrak{A}$ intersecting when exists c such that $latex c\ne 0$ and $latex c\subseteq a$ and $latex c\subseteq b$. […]
Funcoids are more important than topological spaces
I think funcoids are more important for mathematics than topological spaces. Why I think so? Because funcoids have “smoother” (more beautiful) properties than topological spaces. Funcoids were discovered by me. Does the author mean that his discovery of funcoids was more important than the discovery of topological spaces? No. Either topological spaces or funcoids are […]
Pointfree funcoids – a category
I updated the draft of my article “Pointfree Funcoids” at my Algebraic General Topology site. The new version of the article defines pointfree funcoids differently than before: Now a pointfree funcoid may have different posets as its source and destination. So pointfree funcoids now form a category whose objects are posets with least element and […]
Generalization in ZF
I wrote short article “Generalization in ZF” accompanied with Isabelle/ZF sources. This is a draft and alpha. I await your comments on both the article and Isabelle sources. I’m sure my Isabelle sources may be substantially improved (and I plan to work over this). Comments are welcome. After hearing your comments and improving the files, […]
A little error corrected
I corrected a small error in “Filters on Posets and Generalizations” article. The error was in Appendix B in the proof of the theorem stating $latex (t;x)\not\in S$ (I messed $latex t$ and $latex \{t\}$.)
Discrete funcoid which is not complemented
I found an example of a discrete funcoid which is not a complemented element of the lattice of funcoids. Thus the set of discrete funcoids is not the center of the lattice of funcoids, as I conjectured earlier. See the Appendix “Some counter-examples” in my article “Funcoids and Reloids”. The example is simple diagonal relation.