**Theorem**
, , ,
for every composable morphisms , of a category with restricted identities.

**Proof**
and it implies
. The rest follows easily.

**Corollary**
, whenever / are defined.

I don’t think this is essential. The proofs are not the most important thing in my book. The most essential thing are definitions. The proofs are just to fill the gaps. So I deem it not important whether proofs are motivated.

Also, note “algebraic” in the title of my book. The proofs are meant to be just sequences of formulas for as much as possible It is to be thought algebraically. The meaning are the formulas themselves.

Maybe it makes sense to read my book skipping all the proofs? Some proofs contain important ideas, but most don’t. The important thing are definitions.

]]>Unfixed morphisms is a tool for turning a category (with certain extra structure) into a semigroup, that is abstracting away objects.

Currently this research is available in this draft.

I am going to rewrite my online book using unfixed morphisms.

I will add to my book the definition and basic properties of unfixed morphisms. Then I will describe the most basic properties of funcoids and reloids (see my book) necessary to define unfixed funcoids and unfixed reloids. Then I will rewrite my theory of funcoids and reloids into the little more general theory of unfixed funcoids and unfixed reloids.

So I plan to rewrite my entire book.

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

The old version is preserved in Git history.

]]>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.)

]]>Basically, a category with restricted identities is a category together with morphisms which are strictly less (in our order of morphisms) than identities . 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 ” instead of “subset of our object “, that is it allows not to mention the objects for which the sets or filters are considered.

]]>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).

]]>