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September 16, 2014 / Victor Porton

Two new theorems about complete reloids

I have just proven the following two new theorems:

Theorem Composition of complete reloids is complete.

Theorem (\mathsf{RLD})_{\mathrm{out}} g \circ (\mathsf{RLD})_{\mathrm{out}} f = (\mathsf{RLD})_{\mathrm{out}} (g \circ f) if f and g are both complete funcoids (or both co-complete).

See this note for the proofs.

September 16, 2014 / Victor Porton

Another (easy) new theorem

I’ve proved this conjecture (not a long standing conjecture, it took just one day to solve it) and found a stronger theorem than these propositions.

So my new theorem:

Theorem (\mathsf{FCD}) and (\mathsf{RLD})_{\mathrm{out}} form mutually inverse bijections between complete reloids and complete funcoids.

For a proof see this note.

September 16, 2014 / Victor Porton

A proposition about complete funcoids and reloids

In this recent blog post I have formulated the conjecture:

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

This conjecture has not been living long, I have quickly proved it in this note.

September 15, 2014 / Victor 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.

September 14, 2014 / Victor Porton

Some new theorems

I’ve proved some new theorems. The proofs are currently available in this PDF file.

Theorem The set of funcoids is with separable core.

Theorem The set of funcoids is with co-separable core.

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\} .

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\} .

It seems (I have not yet checked) that the following conjecture follows from the last theorem:

Conjecture Composition of complete reloids is complete.

September 11, 2014 / Victor Porton

Funcoids are filters conjecture – finally solved

I have published online a short article saying that the set of funcoids is isomorphic to the set of filters on a certain lattice.

Then I found a counter-example and decided that my theorem was wrong. I was somehow sad about this.

But now I’ve realized that the counter-example is wrong.

So we can celebrate my new theorem. Now there is no reason to assume that it is false.

I will add more details to the article and keep you informed about it.

September 9, 2014 / Victor Porton

Funcoids are filters? Conjecture II

Earlier I have conjectured that the set of funcoids is order-isomorphic to the set of filters on the set of finite joins of funcoidal products of two principal filters. For an equivalent open problem I found a counterexample.

Now I propose another similar but weaker open problem:

Conjecture Let U be a set. The set of funcoids on U is order-isomorphic to the set of filters on the set \Gamma (moreover the isomorphism is (possibly infinite) meet of the filter), where \Gamma is the set of unions \bigcup_{X\in S}(X\times Y_X) where S is a finite partition of U and Y\in \mathscr{P} U for every X\in S

The last conjecture is equivalent to this question formulated in elementary terms. If you solve this (elementary) problem, it could be a major advance in mathematics.

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