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October 17, 2018 / porton

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.

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October 15, 2018 / porton

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.

October 13, 2018 / porton

Equivalence filters and the lattice of filters on the poset of small sets

Equivalence of filters can be described in terms of the lattice of filters on the poset of small sets.

See new subsection “Embedding into the lattice of filters on small” in my book.

October 13, 2018 / porton

Poset of unfixed filters

I defined partial order on the set of unfixed filters and researched basic properties of this poset.

Also some minor results about equivalence of filters.

See my book.

October 11, 2018 / porton

Rebase of filters and equivalent filters

I added a few new propositions into “Equivalent filters and rebase of filters” section (now located in the chapter “Filters and filtrators”) of my book.

I also define “unfixed filters” as the equivalence classes of (small) filters on sets.

This is a step forward to also define “unfixed funcoids” and “unfixed reloids”.

October 9, 2018 / porton

Double filtrators (new book section)

I’ve added a new section “Double filtrators” to the book “Algebraic General Topology. Volume 1”.

I show that it’s possible to describe (\mathsf{FCD}), (\mathsf{RLD})_{\mathrm{out}}, and (\mathsf{RLD})_{\mathrm{in}} entirely in terms of filtrators (order). This seems not to lead to really interesting results but it’s curious.

July 14, 2018 / porton

New easy theorem

I have added a new easy (but unnoticed before) theorem to my book:

Proposition (\mathsf{RLD})_{\mathrm{out}} f\sqcup (\mathsf{RLD})_{\mathrm{out}} g = (\mathsf{RLD})_{\mathrm{out}}(f\sqcup g) for funcoids f, g.