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 $latex f$.
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.
A monovalued reloid with atomic domain is atomic
In the last revision of Funcoids and Reloids online article I proved that every monovalued reloid with atomic domain is atomic. Consequently two following conjectures are proved true: Conjecture A monovalued reloid restricted to an atomic filter object is atomic or empty. Conjecture A (monovalued) function restricted to an atomic filter object is atomic or […]
A counter-example for a conjecture
In a new edition of Funcoids and Reloids article (section “Some counter-examples”) I wrote a counter-example against this conjecture, upholding that there exists a reloid with atomic domain, which is neither injective nor constant. The conjecture is equivalent to this my MathOverflow question, which was quickly solved by my colleagues. I just adapted the proof […]
Two new conjectures in “Funcoids and Reloids” article
Though my Funcoids and Reloids article was declared as a preprint candidate, I made a substantial addendum to it: Added definitions of injective, surjective, and bijective morphisms. Added a conjecture about expressing composition of reloids through atomic reloids. Added a conjecture characterizing monovalued reloids with atomic domains.
Characterization of monovalued reloids with atomic domains
Conjecture Every monovalued reloid with atomic domain is either an injective reloid; a restriction of a constant function (or both).