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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 […]

Restricting a reloid to a trivial atomic filter object

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.

If a reloid is both complete and co-complete it is discrete

Today I proved the following conjecture: If a reloid is both complete and co-complete it is discrete. The proof was easily constructed by me shortly after I noticed an obvious but not noticed before proposition: Proposition A reloid $latex f$ is complete iff there exists a function $latex G : \mho \rightarrow \mathfrak{F}$ such that […]

Two new propositions about funcoids and reloids

I proved the following equality for $latex \mathcal{A}$ being a filter object and $latex f$ being either a funcoid: $latex \mathrm{dom}f|^{\mathsf{FCD}}_{\mathcal{A}} = \mathcal{A} \cap^{\mathfrak{F}} \mathrm{dom}f$ or reloid: $latex \mathrm{dom}f|^{\mathsf{RLD}}_{\mathcal{A}} = \mathcal{A} \cap^{\mathfrak{F}} \mathrm{dom}f$. See the updated version of Funcoids and Reloids online article. The proofs are short but they may be not as obvious at […]

My math sickness

I’m sick with the following: I repeatedly formulate conjectures which have trivial counterexamples and am stuck attempting to prove these true not seeing counterexamples. I should less rely on my ideas what is a true conjecture. I just need to become humbler and less proud. Hopefully now I have enough counter-examples for me to follow […]