Restricting a reloid to a trivial atomic filter object
I proved the following (not very hard) theorem: Theorem $latex f|^{\mathsf{RLD}}_{\{ \alpha \}} = \{ \alpha \} \times^{\mathsf{RLD}} \mathrm{im} \left( f|^{\mathsf{RLD}}_{\{ \alpha \}} \right)$ for every reloid $latex f$ and $latex \alpha \in \mho$. See the online article about funcoids and reloids.
If a reloid is both complete and co-complete it is discrete
Today I proved the following conjecture: If a reloid is both complete and co-complete it is discrete. The proof was easily constructed by me shortly after I noticed an obvious but not noticed before proposition: Proposition A reloid $latex f$ is complete iff there exists a function $latex G : \mho \rightarrow \mathfrak{F}$ such that […]
Two new propositions about funcoids and reloids
I proved the following equality for $latex \mathcal{A}$ being a filter object and $latex f$ being either a funcoid: $latex \mathrm{dom}f|^{\mathsf{FCD}}_{\mathcal{A}} = \mathcal{A} \cap^{\mathfrak{F}} \mathrm{dom}f$ or reloid: $latex \mathrm{dom}f|^{\mathsf{RLD}}_{\mathcal{A}} = \mathcal{A} \cap^{\mathfrak{F}} \mathrm{dom}f$. See the updated version of Funcoids and Reloids online article. The proofs are short but they may be not as obvious at […]
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 […]