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.
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).
Conjecture: Composition of reloids through atomic reloids
The following is a new conjecture: If $latex f$ and $latex g$ are reloids, then $latex g \circ f = \bigcup{}^{\mathsf{RLD}} \{G \circ F | F \in \mathrm{atoms}^{\mathsf{RLD}} f, G \in \mathrm{atoms}^{\mathsf{RLD}} g \}$.
Funcoids and Reloids updated
I updated Funcoids and Reloids article (recently proposed to be the preprint) correcting small errors.