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.
Funcoids and Reloids updated
I updated the the online draft of “Funcoids and Reloids” article. This is almost ready preprint (which I will be able to submit after I will have “Filters on Posets and Generalizations” published). The most notable change in this edition is corrected an error in the proof of the theorem which characterizes monovaluedness of funcoids. […]
Two open problems about completion of funcoids and reloids
In Funcoids and Reloids online article I added two new open problems: 1. $latex \mathrm{Compl} f = f \setminus^{\ast \mathsf{FCD}} (\Omega\times^{\mathsf{FCD}} \mho)$ for every funcoid $latex f$? 2. $latex \mathrm{Compl} f = f \setminus^{\ast \mathsf{RLD}} (\Omega\times^{\mathsf{RLD}} \mho)$ for every reloid $latex f$?