Conjecture: Composition of reloids through atomic reloids
The following is a new conjecture: If $latex f$ and $latex g$ are reloids, then $latex g \circ f = \bigcup{}^{\mathsf{RLD}} \{G \circ F | F \in \mathrm{atoms}^{\mathsf{RLD}} f, G \in \mathrm{atoms}^{\mathsf{RLD}} g \}$.
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 $latex f$?
Funcoids and Reloids – updated
My online article Funcoids and Reloids as well as my list of open problems are updated. Added two open problems: 1. $latex (\mathsf{RLD})_{\mathrm{in}}$ is not a lower adjoint (in general)? 2. $latex (\mathsf{RLD})_{\mathrm{out}}$ is neither a lower adjoint nor an upper adjoint (in general)? Also added an example proving that “$latex (\mathsf{FCD})$ does not preserve […]
Two new open problems about relationships of funcoids and reloids
In the past I overlooked the following two open problems considering them obvious. When I tried to write proofs of these statements down I noticed these are not trivial. So I added them to my list of open problems. Question $latex (\mathsf{RLD})_{\mathrm{out}} ( \mathcal{A} \times^{\mathsf{FCD}} \mathcal{B}) = \mathcal{A} \times^{\mathsf{RLD}} \mathcal{B}$ for every filter objects $latex […]
A counter-example against a distributivity law for funcoids
Example There exist funcoids $latex {f}&fg=000000$ and $latex {g}&fg=000000$ such that $latex \displaystyle ( \mathsf{RLD})_{\mathrm{out}} (g \circ f) \neq ( \mathsf{RLD})_{\mathrm{out}} g \circ ( \mathsf{RLD})_{\mathrm{out}} f. &fg=000000$ Proof: Take $latex {f = {( =)} |_{\Omega}}&fg=000000$ and $latex {g = \mho \times^{\mathsf{FCD}} \left\{ \alpha \right\}}&fg=000000$ for some $latex {\alpha \in \mho}&fg=000000$. Then $latex {( \mathsf{RLD})_{\mathrm{out}} f […]
How connectedness is related with continuity?
I sent my article Connectors and generalized connectedness to a math journal for peer review and publication. This my article however does not address an important facet: It is well known that a set is connected if every function from it to a discrete space is constant. AFAIK, this holds for topological connectedness and continuity, […]
Draft about connectedness
I put online a preliminary draft of my article Connectors and generalized connectedness. This preliminary draft is yet unreadable. The primary reason why I put this unreadable preliminary draft online is that I want to refer to it from this MathOverflow question. From the article: This article is an unreadable preliminary draft. Some definitions and […]
A theorem generalized
I generalized a theorem in the preprint article “Filters on posets and generalizations” on my Algebraic General Topology site. The new theorem is formulated as following: Theorem If $latex (\mathfrak{A}; \mathfrak{Z})$ is a join-closed filtrator and $latex \mathfrak{A}$ is a meet-semilattice and $latex \mathfrak{Z}$ is a complete lattice, then $latex \mathrm{Cor}’ (a \cap^{\mathfrak{A}} b) = […]
Erroneous theorem became a conjecture
I mistakenly used yet unproved statement that $latex \mathrm{up}\,f$ (taken on the filtrator of funcoids) is a filter for every funcoid $latex f$ in proof of a theorem. So after I found this error I downgrade this theorem to the status of conjecture: Conjecture $latex (\mathsf{FCD}) (\mathsf{RLD})_{\mathrm{out}} f = f$ for every funcoid $latex f$.
False proof of an open problem
I earlier proclaimed that I positively solved this conjecture: Conjecture $latex f\cap^{\mathsf{FCD}} g = f\cap g$ for every binary relations $latex f$ and $latex g$. There were error in my proof and I deleted it. See the draft of Funcoids and Reloids article for details on the current state of the problem.