The below is wrong! The proof requires $\langle g^{-1}\rangle J$ 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 $g\circ f$ to be complete it’s enough $f$ to be complete.

The proof which I missed for years is rather trivial:

$\bigsqcup S \mathrel{[g \circ f]} J \Leftrightarrow J \mathrel{[f^{- 1} \circ g^{- 1}]} \bigsqcup S \Leftrightarrow \langle g^{- 1} \rangle J \mathrel{[f^{- 1}]} \bigsqcup S \Leftrightarrow \bigsqcup S \mathrel{[f]} \langle g^{- 1} \rangle J \Leftrightarrow \exists \mathcal{I} \in S : \mathcal{I} \mathrel{[f]} \langle g^{- 1} \rangle J \Leftrightarrow \exists \mathcal{I} \in S : \mathcal{I} \mathrel{[g \circ f]} J$

Thus $g\circ f$ 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 $f \in \mathsf{FCD} (A, B)$, a finite set $X \in \mathscr{P} A$ and a function $t \in \mathscr{F} (B)^X$ there exists (obviously unique) $g \in \mathsf{FCD} (A, B)$ such that $\langle g\rangle p = \langle f \rangle p$ for $p \in \mathrm{atoms}^{\mathscr{F} (A)} \setminus \mathrm{atoms}\, X$ and $\langle g\rangle @\{ x \} = t (x)$ for $x \in X$.

This funcoid $g$ is determined by the formula

$g = (f \setminus (@X \times^{\mathsf{FCD}} \top)) \sqcup \bigsqcup_{x \in X} (@\{ x \} \times^{\mathsf{FCD}} t (x)) .$

and its corollary:

Corollary If $f \in \mathsf{FCD} (A, B)$, $x \in A$, and $\mathcal{Y} \in \mathscr{F} (B)$, then there exists an (obviously unique) $g \in \mathsf{FCD} (A, B)$ such that $\langle g\rangle p = \langle f \rangle p$ for all ultrafilters $p$ except of $p = @\{ x \}$ and $\langle g \rangle @\{ x \} = \mathcal{Y}$.

This funcoid $g$ is determined by the formula

$g = (f \setminus (@\{ x \} \times^{\mathsf{FCD}} \top)) \sqcup (\{ x \} \times^{\mathsf{FCD}} \mathcal{Y})$.

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:

$f=\bot$ and $z(p)=\top$ for infinite sets $A$ and $B$.

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 $f \in \mathsf{FCD} (A ; B)$ and $z \in \mathscr{F} (B)^A$. Then there is an (obviously unique) funcoid $g \in \mathsf{FCD} (A ; B)$ such that $\langle g\rangle x = \langle f\rangle x$ for nontrivial ultrafilters $x$ and $\langle g\rangle @\{ p \} = z (p)$ for $p \in A$

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

I proved the following (in)equalities, solving my open problem which stood for a few months:

$\lvert \mathbb{R} \rvert_{>} \sqsubset \lvert \mathbb{R} \rvert_{\geq} \sqcap \mathord{>}$

$\lvert \mathbb{R} \rvert_{>} = \lvert \mathbb{R} \rvert_{>} \sqcap \mathord{>}$

The proof is currently available in the section “Some inequalities” of this PDF file.

Note that earlier I put online some erroneous proof related to this.

I have published What is physical reality? blog post in my other blog. The post is philosophical.