A new easy theorem in my draft.

Theorem $\mathrm{DOM} (g \circ f) \supseteq \mathrm{DOM} f$, $\mathrm{IM} (g \circ f) \supseteq \mathrm{IM} g$, $\mathrm{Dom} (g \circ f) \supseteq \mathrm{Dom} f$, $\mathrm{Im} (g \circ f) \supseteq \mathrm{Im} g$ for every composable morphisms $f$, $g$ of a category with restricted identities.

Proof $\mathcal{E}_{\mathcal{C}}^{Y, \mathrm{Dst} f} \circ \mathcal{E}_{\mathcal{C}}^{\mathrm{Dst} f, Y} \circ g \circ f = g \circ f \Leftarrow \mathcal{E}_{\mathcal{C}}^{Y, \mathrm{Dst} f} \circ \mathcal{E}_{\mathcal{C}}^{\mathrm{Dst} f, Y} \circ g = g$ and it implies $\mathrm{IM} (g \circ f) \supseteq \mathrm{IM} g$. The rest follows easily.

Corollary $\mathrm{dom} (g \circ f) \sqsubseteq \mathrm{dom} f$, $\mathrm{im} (g \circ f) \sqsubseteq \mathrm{im} g$ whenever $\mathrm{dom}$/$\mathrm{im}$ are defined.

This is a very short addition to my book, in response to a person who criticized my style. He may be partly right, but:

The proofs are generally hard to follow and unpleasant to read as they are just a bunch of equations thrown at you, without motivation or underlying reasoning, etc.

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.

I have rewritten the section “More results on restricted identities” of this draft. Now it contains some new (easy but important) formulas.

I have developed my little addition to category theory, definition and research of properties of unfixed morphisms.

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.

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.