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 $latex \mathcal{A}$.
A conjecture proved
A proof of the following conjecture (now a theorem) was quickly found by me after its formulation: Theorem $latex \left\langle f \right\rangle \mathcal{X} = \bigcap^{\mathfrak{F}} \left\{ \left\langle F \right\rangle \mathcal{X} | F \in \mathrm{up}f \right\}$ for every funcoid $latex f$ and f.o. $latex \mathcal{X}$. See the updated version of my article “Funcoids and Reloids” for […]
Isomorphism of filters expressed through reloids
In the new updated version of the article “Funcoids and Reloids” I proved the following theorem: Theorem Filter objects $latex \mathcal{A}$ and $latex \mathcal{B}$ are isomorphic iff exists a monovalued injective reloid $latex f$ such that $latex \mathrm{dom}f = \mathcal{A}$ and $latex \mathrm{im}f = \mathcal{B}$.
Changes in “Funcoids and Reloids”
I added one new proposition and two open problems to my online article “Funcoids and Reloids”: Conjecture $latex \left\langle f \right\rangle \mathcal{X} = \bigcap^{\mathfrak{F}} \left\{ \left\langle F \right\rangle \mathcal{X} | F \in \mathrm{up}f \right\}$ for every funcoid $latex f$ and f.o. $latex \mathcal{X}$. Proposition $latex \mathrm{dom}( \mathsf{\mathrm{FCD}}) f =\mathrm{dom}f$ and $latex \mathrm{im}(\mathsf{\mathrm{FCD}}) f =\mathrm{im}f$ for […]
Counter-examples against two conjectures
I added counter-examples to the following two conjectures to my online article “Funcoids and Reloids”: Conjecture $latex (\mathsf{RLD})_{\mathrm{out}}(\mathcal{A}\times^{\mathsf{FCD}}\mathcal{B})=\mathcal{A}\times^{\mathsf{RLD}}\mathcal{B}$ for every filter objects $latex \mathcal{A}$ and $latex \mathcal{B}$. Conjecture $latex (\mathsf{RLD})_{\mathrm{out}}(\mathsf{FCD})f=f$ for every reloid $latex f$.