Categories with restricted identities
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 $latex \mathcal{C}$ together with morphisms $latex \mathrm{id}^{\mathcal{C}(A,B)}_X$ […]
“Unfixed filters” research now in the book volume-1
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).
Filters on a lattice are a lattice
I’ve proved that filters on a lattice are a lattice. See my book.
I strengthened a theorem
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.
Math volunteer job
I welcome you to the following math research volunteer job: Participate in writing my math research book (volumes 1 and 2), a groundbreaking general topology research published in the form of a freely downloadable book: implement existing ideas, propose new ideas develop new theories solve open problems write and rewrite the book and other files […]
An error in my book
I erroneously concluded (section “Distributivity of the Lattice of Filters” of my book) that the base of every primary filtrator over a distributive lattice which is an ideal base is a co-frame. Really it can be not a complete lattice, as in the example of the lattice of the poset of all small (belonging to […]
A theorem with a diagram about unfixed filters
I’ve added to my book a theorem with a triangular diagram of isomorphisms about representing filters on a set as unfixed filters or as filters on the poset of all small (belonging to a Grothendieck universe) sets. The theorem is in the subsection “The diagram for unfixed filters”.
Error in my book
There is an error in recently added section “Equivalent filters and rebase of filters” of my math book. I uploaded a new version of the book with red font error notice. The error seems not to be serious, however. I think all this can be corrected. Other sections of the book are not affected at […]
New important theorem
I have proved that \mathscr{S} is an order isomorphism from the poset of unfixed filters to the poset of filters on the poset of small sets. This reveals the importance of the poset of filters on the poset of small sets. See the new version of my book.
New version of my math book (“Equivalent filters and rebase of filters” section)
I have added to my book, section “Equivalent filters and rebase of filters” some new results. Particularly I added “Rebase of unfixed filters” subsection. It remains to research the properties of the lattice of unfixed filters.