I added to my online research book the following theorem:

**Theorem** Let be a distributive lattice with least element. Let . If exists, then also exists and .

The user quasi of Math.SE has helped me with the proof.

I’ve published in my blog a new theorem.

The proof was with an error (see the previous edited post)!

**The below is wrong!** The proof requires to be a principal filter what does not necessarily hold.

I knew that composition of two complete funcoids is complete. But now I’ve found that for to be complete it’s enough to be complete.

The proof which I missed for years is rather trivial:

Thus is complete.

I will amend my book when (sic!) it will be complete.

I have proved (see new version of my book) the following proposition. (It is basically a special case of my erroneous theorem which I proposed earlier.)

**Proposition** For , a finite set and a function there exists (obviously unique) such that for and for .

This funcoid is determined by the formula

and its corollary:

**Corollary** If , , and , then there exists an (obviously unique) such that for all ultrafilters except of and .

This funcoid is determined by the formula

I found the exact error noticed in Error in my theorem post.

The error was that I claimed that infimum of a greater set is greater (while in reality it’s lesser).

I will delete the erroneous theorem from my book soon.

It seems that there is an error in proof of this theorem.

Alleged counter-example:

and for infinite sets and .

I am now attempting to locate the error in the proof.

I have proved (and added to my online book) the following theorem:

**Theorem** Let and . Then there is an (obviously unique) funcoid such that for nontrivial ultrafilters and for

After I started to prove it, it took about a hour or like this to finish the proof.

0