# 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.

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