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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.

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.

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.