I proved the following two elementary but useful theorems: Theorem For every funcoids $latex f$, $latex g$: If $latex \mathrm{im}\, f \supseteq \mathrm{im}\, g$ then $latex \mathrm{im}\, (g\circ f) = \mathrm{im}\, g$. If $latex \mathrm{im}\, f \subseteq \mathrm{im}\, g$ then $latex \mathrm{dom}\,…

read moreI proved the following simple theorem: 1. $latex (\mathsf{FCD}) f$ is a monovalued funcoid if $latex f$ is a monovalued reloid. 2. $latex (\mathsf{FCD}) f$ is an injective funcoid if $latex f$ is an injective reloid. See online article “Funcoids and Reloids”…

read moreTo the theorem “Every monovalued reloid is a restricted function.” I added a new corollary “Every monovalued injective reloid is a restricted injection.” See the online article “Funcoids and Reloids” and this Web page.

read moreI updated online article “Pointfree Funcoids” from “preliminary draft” to just “draft”. This means that it was somehow checked for errors and ready for you to read. (No 100% warranty against errors however.) Having finished with that draft the way is now…

read moreMy draft article Pointfree Funcoids was not yet thoroughly checked for errors. However at this stage of the draft I expect that there are no big errors there only possible little errors. Familiarize yourself with Algebraic General Topology.

read moreI proved true the following conjecture: Theorem $latex \mathrm{Compl} \left( \bigcup^{\mathsf{RLD}} R \right) = \bigcup ^{\mathsf{RLD}} \langle \mathrm{Compl} \rangle R$ for every set $latex R$ of reloids. The following conjecture remains open: Conjecture $latex \mathrm{Compl}\,f \cap^{\mathsf{RLD}} \mathrm{Compl}\,g =\mathrm{Compl} (f \cap^{\mathsf{RLD}} g)$ for…

read moreI proved true the following conjecture: Theorem $latex \mathrm{Compl} \, \mathrm{CoCompl}\, f = \mathrm{CoCompl}\, \mathrm{Compl}\, f = \mathrm{Cor}\, f$ for every reloid $latex f$. See here for definitions and proofs.

read moreI proved the following two simple theorems: Proposition $latex \mathrm{Compl}f = \bigcup^{\mathsf{FCD}} \left\{ f|^{\mathsf{FCD}}_{\{ \alpha \}} \middle| \alpha \in \mho \right\}$ for every funcoid $latex f$. Proposition $latex \mathrm{Compl}f = \bigcup^{\mathsf{RLD}} \left\{ f|^{\mathsf{RLD}}_{\{ \alpha \}} \middle| \alpha \in \mho \right\}$ for every…

read moreI proved the following two similar theorems about funcoids and reloids: Theorem For a complete funcoid $latex f$ there exist exactly one function $latex F \in \mathfrak{F}^{\mho}$ such that $latex f = \bigcup^{\mathsf{FCD}} \left\{ \{ \alpha \} \times^{\mathsf{FCD}} F(\alpha) | \alpha \in…

read moreI proved the following (not very hard) theorem: Theorem $latex f|^{\mathsf{RLD}}_{\{ \alpha \}} = \{ \alpha \} \times^{\mathsf{RLD}} \mathrm{im} \left( f|^{\mathsf{RLD}}_{\{ \alpha \}} \right)$ for every reloid $latex f$ and $latex \alpha \in \mho$. See the online article about funcoids and reloids.

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