I removed from my draft sections about “categories under Rel”.

The removal happened because I developed a more general and more beautiful theory.

The old version is preserved in Git history.

I announced that I have introduces axioms for “restricted identities”, a structure on a category which allows to turn the category into a semigroup (abstracting away objects).

But I noticed that these axioms do not fit into concrete examples which I am going to research.

So I have rewritten the text about restricted identities with somehow different axioms.

The theory of categories with restricted identities is presently presented in this rough draft not yet enough checked for errors. (I am going to move it to the main book after thorough examination.)

“Unfixed filter” section of my book was rewritten for more general lattices instead of old version with a certain lattice of sets.

In this draft (to be moved into the online book in the future, but the draft is nearing finishing this topic, not including functors between categories with restricted identities) I described axioms and properties of categories with restricted identities.

Basically, a category with restricted identities is a category $\mathcal{C}$ together with morphisms $\mathrm{id}^{\mathcal{C}(A,B)}_X$ which are strictly less (in our order of morphisms) than identities $1^A$. These “restricted identities” conform to certain axioms.

Using restricted identities, it is possible to turn a category into a semigroup, which I call “semigroup of unfixed morphisms”, because semigroups elements don’t have “fixed” source and destination objects, but describe common properties of morphisms with different sources and destinations (abstracting objects of the category away).

I wrote all this with the purpose to define “unfixed funcoids” and “unfixed reloids”, to allow abstract away the source and destination of say a funcoid, making it similar to “arbitrary binary relation” instead of limiting to binary relations between two given sets. This increases abstraction and may increase expressiveness. Particularly this allows to use just “set $X$” instead of “subset $X$ of our object $A$“, that is it allows not to mention the objects for which the sets or filters are considered.

I essentially finished my research of unfixed filters.

I moved all research of unfixed filters to volume-1.pdf. Particularly now it contains subsections “The lattice of unfixed filters” and “Principal unfixed filters and filtrator of unfixed filters”.

Now I am going to research unfixed reloids and unfixed funcoids (yet to be defined).

I’ve proved that filters on a lattice are a lattice.

See my book.

I strengthened a theorem: It is easily provable that every atomistic poset is strongly separable (see my book).

It is a trivial result but I had a weaker theorem in my book before today.