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 $latex f$.
Filters on Posets and Generalizations updated
I uploaded updated version of Filters on Posets and Generalizations article and sent it to Armenian Journal of Mathematics for peer review. The main change in this version is a counter-example to the conjecture, that every weak partition of a filter object on a set is a strong partition. This example was suggested me by […]
Question: Complete classification of ultrafilters?
Are there a known complete classification of filters (or at least ultrafilters)? By complete classification I mean a characterization of every filter by a family of cardinal numbers such that two filters are isomorphic if and only if they have the same characterization. For definition of isomorphic filters see my article “Filters on Posets and […]
“Filters on Posets and Generalizations” updated
I updated my online draft of the “Filters on Posets and Generalizations” article, while a former version of it was submitted as a preprint into Armenian Journal of Mathematics. The main new feature of my online draft is the section “Complementive filter objects and factoring by a filter” added and also a counterexample against this […]
Distributivity of composition over join of reloids
In the last version of Reloids and Funcoids online article I proved true the following conjecture: Theorem If $latex f$, $latex g$, $latex h$ are reloids then $latex f\circ (g\cup^{\mathsf{RLD}} h) = f\circ g \cup^{\mathsf{RLD}} f\circ h$; $latex (g\cup^{\mathsf{RLD}} h) \circ f = g\circ f \cup^{\mathsf{RLD}} h\circ f$.
Every monovalued reloid is a restricted function
The conjecture “Every monovalued reloid is a restricted function” is proved true as a corollary of this theorem in the last revision of Funcoids and Reloids online article.