Orderings of filters in terms of reloids – online draft
I put a preliminary draft (not yet checked for errors) of the article “Orderings of filters in terms of reloids” at Algebraic General Topology site. The abstract Orderings of filters which extend Rudin-Keisler preorder of ultrafilters are defined in terms of reloids that (roughly speaking) is filters on sets of binary relations between some sets. […]
Two elementary theorems
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 $latex \mathrm{dom}\, (g\circ f) = \mathrm{dom}\, g$. Theorem For every reloids $latex f$, $latex g$: […]
Funcoid corresponding to a monovalued reloid is monovalued
I 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” and this Web page.
New corollary
To 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.
“Pointfree Funcoids” is now a good draft
I 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 free for my further Algebraic General Topology research, such as n-ary (multidimensional) funcoids, […]
Draft: Pointfree funcoids
My 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.
Completion of a join of reloids
I 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 every reloids $latex f$ and $latex g$. See here for definitions and proofs.
Compl CoCompl f = CoCompl Compl f = Cor f
I 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.
Yet two simple theorems
I 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 reloid $latex f$. See this online article for definitions and proofs.
Two similar theorems about funcoids and reloids
I 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 $latex f = \bigcup^{\mathsf{FCD}} \left\{ \{ \alpha \} \times^{\mathsf{FCD}} F(\alpha) | \alpha \in \mho \right\}$. For a co-complete funcoid $latex f$ there exist exactly one function […]