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 […]
Funcoids and Reloids – updated
My online draft article Funcoids and Reloids updated with minor changes in “Connectedness regarding funcoids and reloids” section.
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 […]
My research plans shifted
Yesterday 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 into a binary relation between n-ary tuples and m-ary tuples). But this funcoid […]
My research plan
I ‘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 about funcoids and reloids (despite of there are yet several open problems in […]
Funcoids and reloids, a Galois connection
I 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.
A counterexample: Funcoid corresponding to outer reloid
I 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 and Reloids online article, the section Some counter-examples for this counterexample.
A counterexample against “Meet of discrete funcoids is discrete”
I 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 $latex f \cap^{\mathsf{FCD}} g = {(=)} |_{\Omega}$ (where $latex \Omega$ is the Frechet filter object). […]
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) = […]