I’ve proved the following (for every funcoids and ):
Statement or equivalently: If then there exists , such that .
But now I’ve noticed that the proof was with an error! So it is again a conjecture.
I’ve proved the following lemma:
Lemma Let for every and there is a such that .
Then for every and there is a such that .
I spent much time (probably a few hours) to prove it, but the found proof is really simple, almost trivial.
The proof is currently located in this PDF file.
After prayer in tongues and going down anointment of Holy Spirit I proved this conjecture about funcoids. The proof is currently located in this PDF file. Well, the proof is for special cases of distributive lattices, but more general case seems not necessary (at least now).
It seems easy to generalize it for more general lattices than the lattice of funcoids, what I hope to do a little later.
Conjecture for all funcoids , (with corresponding sources and destinations).
Looks trivial? But how to (dis)prove it?