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 […]
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.
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 […]
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.
Pointfree funcoids
I put on the Web the first preliminary draft of my article “Pointfree Funcoids”. It seems that pointfree funcoids is a useful tool to research n-ary (multidimensional as opposed to binary) funcoids which in turn is a useful tool to research operations on values of generalized limits of n-ary functions. See Algebraic General Topology for […]
Two propositions and a conjecture
I added to Funcoids and Reloids article the following two new propositions and a conjecture: Proposition $latex (\mathsf{FCD}) (f\cap^{\mathsf{RLD}} ( \mathcal{A}\times^{\mathsf{RLD}} \mathcal{B})) = (\mathsf{FCD}) f \cap^{\mathsf{FCD}} (\mathcal{A}\times^{\mathsf{FCD}} \mathcal{B})$ for every reloid $latex f$ and filter objects $latex \mathcal{A}$ and $latex \mathcal{B}$. Proposition $latex ( \mathsf{RLD})_{\mathrm{in}} (f \cap^{\mathsf{FCD}} ( \mathcal{A} \times^{\mathsf{FCD}} \mathcal{B})) = (( \mathsf{RLD})_{\mathrm{in}} f) […]
Added a new proposition
I added the following proposition to the Funcoids and Reloids article: Proposition $latex (\mathsf{FCD})I_{\mathcal{A}}^{\mathsf{RLD}} = I_{\mathcal{A}}^{\mathsf{FCD}}$ for every filter object $latex \mathcal{A}$.