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September 15, 2014 / porton

Correction on the recent theorems

About new theorems in in this my blog post:

I’ve simplified this theorem:

Theorem A reloid f is complete iff
f = \bigsqcap^{\mathsf{RLD}} \left\{ \bigcup_{x \in \mathrm{Src}\, f}     (\{ x \} \times \langle T \rangle^{\ast} \{ x \}) \, |     \, T \in (\mathscr{P} \mathrm{Dst}\, f)^{\mathrm{Src}\, f}, \forall x \in     A : \langle T \rangle^{\ast} \{ x \} \in G (x) \right\} .

into

Theorem A reloid f is complete iff f=(\mathsf{RLD})_{\mathrm{out}} g for a complete funcoid g.

For a proof see this note.

The next theorem:

Theorem A funcoid f is complete iff
f = \bigsqcap^{\mathsf{FCD}} \left\{ \bigcup_{x \in \mathrm{Src}\, f}     (\{ x \} \times \langle T \rangle^{\ast} \{ x \}) \, |     \, T \in (\mathscr{P} \mathrm{Dst}\, f)^{\mathrm{Src}\, f}, \forall x \in     A : \langle T \rangle^{\ast} \{ x \} \in G (x) \right\} .

collapses into

Theorem f=\bigsqcap^{\mathsf{FCD}} \mathrm{up}\, f, what I proved long time ago.

So, I have removed this theorem from my writings.

Finally, I add the conjecture:

Conjecture A funcoid f is complete iff f=(\mathsf{FCD}) g for a complete reloid g.

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