Complete lattice generated by a partitioning of a lattice element
In this post I defined strong partitioning of an element of a complete lattice. For me it was seeming obvious that the complete lattice generated by the set where is a strong partitioning is equal to . But when I actually tried to write down the proof of this statement I found that it is not obvious to prove. So I present this to you as a conjecture:
Conjecture The complete lattice generated by a strong partitioning of an element of a complete lattice is equal to .
Proposition Provided that this conjecture is true, we can prove that the complete lattice generated by a strong partitioning of an element of a complete lattice is a complete atomic boolean lattice with the set of its atoms being (Note: So is completely distributive).
Proof Completeness of is obvious. Let . Then exists such that . Let . Then and . is the biggest element of . So we have proved that is a boolean lattice.
Now let prove that is atomic with the set of atoms being . Let and . If then either or where , and . Because is a strong partitioning, and . So .
Finally we will prove that elements of are not atoms. Let and . Then where and . If is an atom then what is impossible. QED
The above conjecture as a step to solution to the original conjecture may also be considered for the polymath research problem. Or maybe we should research both these two problems in a single polymath set, as the solution of one of them may inspire the solution of the other of these two problems.