I have just proven the following two new theorems:

**Theorem** Composition of complete reloids is complete.

**Theorem** if and are both complete funcoids (or both co-complete).

See this note for the proofs.

I’ve proved this conjecture (not a long standing conjecture, it took just one day to solve it) and found a stronger theorem than these propositions.

So my new theorem:

**Theorem** and form mutually inverse bijections between complete reloids and complete funcoids.

For a proof see this note.

In this recent blog post I have formulated the conjecture:

**Conjecture** A funcoid is complete iff for a complete reloid .

This conjecture has not been living long, I have quickly proved it in this note.

About new theorems in in this my blog post:

I’ve simplified this theorem:

**Theorem** A reloid is complete iff

.

into

**Theorem** A reloid is complete iff for a complete funcoid .

For a proof see this note.

The next theorem:

**Theorem** A funcoid is complete iff

.

collapses into

**Theorem** , what I proved long time ago.

So, I have removed this theorem from my writings.

Finally, I add the conjecture:

**Conjecture** A funcoid is complete iff for a complete reloid .

I’ve proved some new theorems. The proofs are currently available in this PDF file.

**Theorem** The set of funcoids is with separable core.

**Theorem** The set of funcoids is with co-separable core.

**Theorem** A funcoid is complete iff

.

**Theorem** A reloid is complete iff

.

It seems (I have not yet checked) that the following conjecture follows from the last theorem:

**Conjecture** Composition of complete reloids is complete.

I have published online a short article saying that the set of funcoids is isomorphic to the set of filters on a certain lattice.

Then I found a counter-example and decided that my theorem was wrong. I was somehow sad about this.

But now I’ve realized that the counter-example is wrong.

So we can celebrate my new theorem. Now there is no reason to assume that it is false.

I will add more details to the article and keep you informed about it.

Earlier I have conjectured that the set of funcoids is order-isomorphic to the set of filters on the set of finite joins of funcoidal products of two principal filters. For an equivalent open problem I found a counterexample.

Now I propose another similar but weaker open problem:

**Conjecture** Let be a set. The set of funcoids on is order-isomorphic to the set of filters on the set (moreover the isomorphism is (possibly infinite) meet of the filter), where is the set of unions where is a finite partition of and for every

The last conjecture is equivalent to this question formulated in elementary terms. If you solve this (elementary) problem, it could be a major advance in mathematics.

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