I conjectured certain formula for the complete lattice generated by a strong partitioning of an element of complete lattice. Now I have found a beautiful proof of a weaker statement than this conjecture. (Well, my proof works only in the case of distributive lattices, but the case of non-distributive lattices is outside of my research area.)
Let’s denote where is a strong partitioning an element of the complete lattice . Our conjecture is trivially equivalent to the statement that is closed under arbitrary meets and joins.
That is closed regarding any joins is obvious. To finish proving the conjecture we need to show that is closed under arbitrary meets. In this post I prove weaker result that is closed under finite meets.
I hope this finite case may serve as a model for the general infinite case. However it seems that generalizing it to infinite case is non-trivial.
Theorem Let is a distributive complete lattice and is a strong partitioning of some element of this lattice. Then is closed under finite meets.
Proof Let .
Applying the formula twice we get
But for any exist such that and . So .