I claimed that I have proved the following conjecture: Conjecture $latex \forall H \in \mathrm{up} (g \circ f) \exists F \in \mathrm{up}\, f, G \in \mathrm{up}\, g : H \sqsupseteq G \circ F$ for every composable funcoids $latex f$ and $latex g$….

read moreWARNING: The proof was with an error! I have proved the following theorem: Theorem $latex \forall H \in \mathrm{up} (g \circ f) \exists F \in \mathrm{up}\, f, G \in \mathrm{up}\, g : H \sqsupseteq G \circ F$ for every composable funcoids $latex…

read moreExample There is such a non-symmetric reloid $latex f$ that $latex (\mathsf{FCD})f$ is symmetric. Take $latex f=((\mathsf{RLD})_{\mathrm{in}}(\mathord{=})|_{\mathbb{R}})\sqcap (\mathord{\geq})_{\mathbb{R}}$. I have added this to my online book.

read moreI’ve released my math research book and all supplementary materials free with semicolons replaced with commas to denote tuples: $latex (a;b)$ → $latex (a,b)$, in order to comply with usual math notation of other mathematicians.

read moreCharacterize the set $latex \{f\in\mathsf{FCD} \mid (\mathsf{RLD})_{\mathrm{in}} f=(\mathsf{RLD})_{\mathrm{out}} f\}$. (This seems a difficult problem.)

read moreI have proved $latex (\mathsf{RLD})_{\mathrm{in}} \Omega^{\mathsf{FCD}} = \Omega^{\mathsf{RLD}}$ (where $latex \Omega^{\mathsf{FCD}}$ is a cofinite funcoid and $latex \Omega^{\mathsf{RLD}}$ is a cofinite reloid that is reloid defined by a cofinite filter). The proof is currently available in this draft. Note that in the…

read moreI’ve found a typo in my math book. I confused existential quantifiers with universal quantifiers in the section “Second product. Oblique product” in the chapter “Counter-examples about funcoids and reloids”.

read moreI added more properties of cofinite funcoids to this draft.

read moreI have described generalized cofinite filters (including the “cofinite funcoid”). See the draft at http://www.math.portonvictor.org/binaries/addons.pdf

read moreDefine for posets with order $latex \sqsubseteq$: $latex \Phi_{\ast} f = \lambda b \in \mathfrak{B}: \bigsqcup \{ x \in \mathfrak{A} \mid f x \sqsubseteq b \}$; $latex \Phi^{\ast} f = \lambda b \in \mathfrak{A}: \bigsqcap \{ x \in \mathfrak{B} \mid f x…

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