Conjecture The following are equivalent (for every lattice of funcoids between some sets and a set of principal funcoids (=binary relations)):
- (for every natural ).
- There exists a funcoid such that .
and are obvious.
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Conjecture Let be a set of binary relations. If for every we have then there exists a funcoid such that .
Definition A set of binary relations is a base of a funcoid when all elements of are above and .
It was easy to show:
Proposition A set of binary relations is a base of a funcoid iff it is a base of .
Today I’ve proved the following important theorem:
Theorem If is a filter base on the set of funcoids then is a base of .
The proof is currently located in this PDF file.
It is yet unknown whether the converse theorem holds, that is whether every base of a funcoid is a filter base on the set of funcoids.
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.