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…
read moreMy online draft article Funcoids and Reloids updated with minor changes in “Connectedness regarding funcoids and reloids” section.
read moreIn 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}}…
read moreExample 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\{…
read moreYesterday I wrote that I next thing which I will research are n-ary funcoids and n-ary reloids. It seems that (n+m)-ary funcoid can be split into a funcoid acting from n-ary funcoids to m-ary funcoids (similarly to (n+m)-ary relation can be split…
read moreI ‘ve said that I take a vacation in my math research work in order to write a religious book. Unexpectedly quickly I have already finished to write and publish this book and return to my mathematical research. Now having researched enough…
read moreI proved that $latex (\mathsf{FCD})$ is the lower adjoint of $latex (\mathsf{RLD})_{\mathrm{in}}$. Also from this follows that $latex (\mathsf{FCD})$ preserves all suprema and $latex (\mathsf{RLD})_{\mathrm{in}}$ preserves all infima. See Algebraic General Topology and specifically Funcoids and Reloids online article.
read moreI found a counter-example to the following conjecture. Conjecture $latex (\mathsf{FCD}) (\mathsf{RLD})_{\mathrm{out}} f = f$ for every funcoid $latex f$. The counterexample is $latex f = {(=)}|_{\Omega}$ where $latex \Omega$ is the Fréchet filter. See Algebraic General Topology and in particular Funcoids…
read moreI found a counterexample to the following conjecture: Conjecture $latex f\cap^{\mathsf{FCD}} g = f\cap g$ for every binary relations $latex f$ and $latex g$. The counter-example is $latex f = {(=)}|_{\mho}$ and $latex g = \mho\times\mho \setminus f$. I proved $latex f…
read moreI 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…
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