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July 13, 2018 / porton

A step forward to solve an open problem

I am attempting to find the value of the node “other” in a diagram currently located at this file, chapter “Extending Galois connections between funcoids and reloids”.

By definition \mathrm{other} = \Phi_{\ast}(\mathsf{RLD})_{\mathrm{out}}.

A few minutes ago I’ve proved (\Phi_{\ast}(\mathsf{RLD})_{\mathrm{out}})\bot = \Omega^{\mathsf{FCD}}, that is found the value of the function “other” at \bot. It is yet a “mistery” what values it has at different arguments.

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July 2, 2018 / porton

New short chapter

I’ve added a new short chapter “Generalized Cofinite Filters” to my book.

June 30, 2018 / porton

A conjecture proved

I have proved the conjecture that S^{\ast}(\mu)\circ S^{\ast}(\mu)=S^{\ast}(\mu) for every endoreloid \mu. The easy proof is currently available in this file.

June 28, 2018 / porton

New theorem about relationships between funcoids and reloids

I have proved (the proof is currently available in this file) that ((\mathsf{FCD}), (\mathsf{RLD})_{\mathrm{in}}) are components of a pointfree funcoid between boolean lattices.

See my book for definitions.

June 25, 2018 / porton

My math book updated

I have updated my math book with new (easy but) general theorem similar to this (but in other notation):

Theorem If \mathfrak{Z} is an ideal base, then the set of filters on \mathfrak{Z} is a join-semilattice and the binary join of filters is described by the formula \mathcal{A}\sqcup\mathcal{B} = \mathcal{A}\cap\mathcal{B}.

I have updated some other theorems to use this general result and so themselves to become a little more general.

In the course of rewriting my book I found and corrected several small errors.

The latest changes of the book are not yet as thoroughly checked for errors as the rest of the book.

June 19, 2018 / porton

My math book updated

I updated my math research book to use “weakly down-aligned” and “weakly up-aligned” instead of “down-aligned” and “up-aligned” (see the book for the definitions) where appropriate to make theorems slightly more general.

During this I also corrected an error. (One theorem referred to complement of a lattice element without stating that the lattice is boolean.)

Well, maybe I introduced new errors. The current version is not 100% stable. However, I am sure the errors (if any) are small and don’t break the exposition.

May 25, 2018 / porton

I proved that certain functors between topological spaces and endofuncoid are adjoint

I proved:

Theorem T is a left adjoint of both F_{\star} and F^{\star}, with bijection which preserves the “function” part of the morphism.

The details and the proof is available in the draft of second volume of my online book.

The proof is not yet enough checked for errors.