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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