I have defined two new kinds of products of funcoids:

1. $\prod^{\mathrm{in}}_{i \in \mathrm{dom}\, f} f = \prod^{(C)}_{i \in \mathrm{dom}\, f} (\mathsf{RLD})_{\mathrm{in}} f_i$ (cross-inner product).
2. $\prod^{\mathrm{out}}_{i \in \mathrm{dom}\, f} f = \prod^{(C)}_{i \in \mathrm{dom}\, f} (\mathsf{RLD})_{\mathrm{out}} f_i$ (cross-outer product).

These products are notable that their values are also funcoids (not just pointfree funcoids).

See new version of my book for details.

I’ve added to my book the following conjecture:

Conjecture For every composable funcoids $f$ and $g$

$(\mathsf{RLD})_{\mathrm{out}}(g\circ f)\sqsupseteq (\mathsf{RLD})_{\mathrm{out}}g\circ(\mathsf{RLD})_{\mathrm{out}} f.$

After noticing an error in my math book, I rewritten its section “Funcoids and filters” to reflect that $(\mathsf{RLD})_\Gamma = (\mathsf{RLD})_{\mathrm{in}}$.

Previously I proved an example demonstrating that $(\mathsf{RLD})_\Gamma \ne (\mathsf{RLD})_{\mathrm{in}}$, but this example is believed by me to be wrong. The example was removed from the book.

Thus I removed all references to $(\mathsf{RLD})_\Gamma$ (as it is the same as $(\mathsf{RLD})_{\mathrm{in}}$) and reworked the chapter “Funcoids and filters” to reflect the change.

The book is available free of change at this Web page.

The story of the past:

$(\mathsf{RLD})_\Gamma$ was defined by the formula $(\mathsf{RLD})_\Gamma f = \bigsqcap^{\mathsf{RLD}} \mathrm{up}^\Gamma\, f$.

From the theorem in “The diagram” section (the theorem with a diagram) it trivially follows that $(\mathsf{RLD})_\Gamma f = (\mathsf{RLD})_{\mathrm{in}} f$. It follows trivially, but I have found this only today.

I proved both $(\mathsf{RLD})_\Gamma \ne (\mathsf{RLD})_{\mathrm{in}}$ and $(\mathsf{RLD})_\Gamma = (\mathsf{RLD})_{\mathrm{in}}$.

So there is an error in my math research book.

I will post the details of the resolution as soon as I will locate and correct the error. While the error is not yet corrected I have added a red font note in my book.

I claimed earlier that I partially solved this open problem.

Today I solved it completely. The proof is available in this PDF file.

I’ve added chapters “Cartesian closedness” and “Singularities” (from the site http://tiddlyspace.com which will be closed soon) to volume 2 draft.

Both chapters are very rough draft and present not rigorous proofs but rough ideas.

The journal European Journal of Pure and Applied Mathematics has accepted my
article after a peer review and asked me to send it in their LaTeX format.

I had a hyperref trouble with my LaTeX file. So I’ve said them that I
withdraw my article.

But later I realized that the best thing I can do is to remove hyperref
package.

After this I successfully converted my article to use their LaTeX package
and send it to European Journal of Pure and Applied Mathematics again. They
then said me that will publish the paper “this year”.

A few years passed.

It is yet unpublished and the editors of the journal ignore my emails, where
I remind them that they agreed to publish my article but don’t publish it.

What to do?

Note that my article is available online (the submitted version has a
different LaTeX template and some minor changes):

http://www.mathematics21.org/binaries/funcoids-reloids.pdf