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Month: March 2011

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Algebraic general topology Filters General Topology

Two elementary theorems

By Victor Porton
On March 30, 2011

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}\,…

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Algebraic general topology General Topology

Funcoid corresponding to a monovalued reloid is monovalued

By Victor Porton
On March 29, 2011

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”…

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Algebraic general topology Filters General Topology

New corollary

By Victor Porton
On March 29, 2011

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.

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Algebraic general topology General Topology Pointfree topology

“Pointfree Funcoids” is now a good draft

By Victor Porton
On March 26, 2011

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…

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Algebraic general topology General Topology Pointfree topology

Draft: Pointfree funcoids

By Victor Porton
On March 23, 2011

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.

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Algebraic general topology General Topology

Completion of a join of reloids

By Victor Porton
On March 18, 2011

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…

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Algebraic general topology Filters General Topology

Compl CoCompl f = CoCompl Compl f = Cor f

By Victor Porton
On March 18, 2011

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.

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#completion#reloids
Algebraic general topology General Topology

Yet two simple theorems

By Victor Porton
On March 17, 2011

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…

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Algebraic general topology General Topology

Two similar theorems about funcoids and reloids

By Victor Porton
On March 17, 2011

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…

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Uncategorized

Restricting a reloid to a trivial atomic filter object

By Victor Porton
On March 17, 2011

I 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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  • SCIENCE
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  • Algebraic General Topology
  • Axiomatic Theory of Formulas
  • Limit of a Discontinuous Function
  • More
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  • SCIENCE
  • Home
    • Blog
    • Soft
  • Algebraic General Topology
    • Paperback
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    • PDF
  • Axiomatic Theory of Formulas
    • Paperback
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  • Limit of a Discontinuous Function
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    • Politics
  • Prize