I’ve introduced another version of cross-composition of funcoids. This forms a category with star-morphisms. It is conjectured that this category is quasi-invertible, because I have failed to prove it. This should be included in the next version of my book.

read moreA mathematician named Todd Trimble has helped me to prove that the set of funcoids between two given sets (and more generally certain pointfree funcoids) is always a co-frame. (I knew this for funcoids but my proof required axiom of choice, while…

read moreI’ve found today earlier stated conjecture that lattices $latex \mathrm{Compl}\mathsf{FCD}(A;B)$ and $latex \mathrm{Compl}\mathsf{RLD}(A;B)$ are co-brouwerian. Exercise: Prove this fact.

read moreI’ve proved the following conjecture: Theorem Let $latex f$ be a staroid such that $latex (\mathrm{form}\, f)_i$ is an atomic lattice for each $latex i \in \mathrm{arity}\, f$. We have $latex \displaystyle L \in \mathrm{GR}\, f \Leftrightarrow \mathrm{GR}\, f \cap \prod_{i \in…

read moreLet $latex \mathfrak{A}$ be an indexed family of sets. Products are $latex \prod A$ for $latex A \in \prod \mathfrak{A}$. Hyperfuncoids are filters $latex \mathfrak{F} \Gamma$ on the lattice $latex \Gamma$ of all finite unions of products. Problem Is $latex \bigsqcap^{\mathsf{FCD}}$ a…

read moreI have completed preliminary error checking for my online article Funcoids are Filters. This article is a major step forward in the theory of funcoids.

read moreWhile walking home from McDonalds I conceived the following idea how we can generalize reloids and funcoids. Let $latex C$ be a category with finite products, the set of objects of which is a complete lattice (for the case of funcoids as…

read moreThin groupoid is an important but a heavily overlooked concept. When I did Google search for “thin groupid” (with quotes), I found just $latex {7}&fg=000000$ (seven) pages (and some of these pages were created by myself). It is very weird that such…

read moreI’ve proved the theorem: Theorem $latex f \mapsto \bigsqcap^{\mathsf{RLD}} f$ and $latex \mathcal{A} \mapsto \Gamma (A ; B) \cap \mathcal{A}$ are mutually inverse bijections between $latex \mathfrak{F} (\Gamma (A ; B))$ and funcoidal reloids. These bijections preserve composition. (The second items is…

read moreTheorem $latex (\mathsf{RLD})_{\mathrm{in}} (g \circ f) = (\mathsf{RLD})_{\mathrm{in}} g \circ (\mathsf{RLD})_{\mathrm{in}} f$ for every composable funcoids $latex f$ and $latex g$. See proof in this online article.

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