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November 27, 2018 / porton

“Unfixed filters” rewritten

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

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November 22, 2018 / porton

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 \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.

October 22, 2018 / porton

“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).

October 22, 2018 / porton

Filters on a lattice are a lattice

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

See my book.

October 21, 2018 / porton

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.

October 19, 2018 / porton

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
  • check for errors
  • help with book LaTeX formatting
  • represent the research at scientific conferences

You can do all of the above or any particular thing, dependently on your mood.

Required skills:

  • basic general topology
  • LaTeX
  • Git

More desired skills:

  • advanced general topology
October 18, 2018 / porton

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 Grothendieck universe) sets.

I will update my book soon.