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 .

A few minutes ago I’ve proved , that is found the value of the function “other” at . It is yet a “mistery” what values it has at different arguments.

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

I have proved the conjecture that for every endoreloid . The easy proof is currently available in this file.

I have proved (the proof is currently available in this file) that are components of a pointfree funcoid between boolean lattices.

See my book for definitions.

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

**Theorem** If is an ideal base, then the set of filters on is a join-semilattice and the binary join of filters is described by the formula .

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.

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.

I proved:

**Theorem** is a left adjoint of both and , 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.

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