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October 24, 2016 / porton

A new little theorem (Galois connections)

I’ve added the following to my research book:

Galois surjection is the special case of Galois connection such that f^{\ast} \circ f_{\ast} is identity.

For Galois surjection \mathfrak{A} \rightarrow \mathfrak{B} such that \mathfrak{A} is a join-semilattice we have (for every y \in \mathfrak{B})

f_{\ast} y = \max \{ x \in \mathfrak{A} \mid f^{\ast} x = y \}.

(Don’t confuse this my little theorem with the well-known theorem with similar formula formula f_{\ast} y = \max \{ x \in \mathfrak{A} \mid f^{\ast} x \leq y \}.)

This formula in particular applies to the Galois connection between funcoids and reloids (see my book).

October 24, 2016 / porton

Online scientific conference – please donate

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September 11, 2016 / porton

Two new kinds of product of funcoids

I have defined two new kinds of products of funcoids:

  1. \prod^{\mathrm{in}}_{i \in \mathrm{dom}\, f} f = \prod^{(C)}_{i \in \mathrm{dom}\, f} (\mathsf{RLD})_{\mathrm{in}} f_i (cross-inner product).
  2. \prod^{\mathrm{out}}_{i \in \mathrm{dom}\, f} f = \prod^{(C)}_{i \in \mathrm{dom}\, f} (\mathsf{RLD})_{\mathrm{out}} f_i (cross-outer product).

These products are notable that their values are also funcoids (not just pointfree funcoids).

See new version of my book for details.

September 10, 2016 / porton

A conjecture about outward funcoids

I’ve added to my book the following conjecture:

Conjecture For every composable funcoids f and g

(\mathsf{RLD})_{\mathrm{out}}(g\circ f)\sqsupseteq (\mathsf{RLD})_{\mathrm{out}}g\circ(\mathsf{RLD})_{\mathrm{out}} f.

September 10, 2016 / porton

An error in my math book corrected

After noticing an error in my math book, I rewritten its section “Funcoids and filters” to reflect that (\mathsf{RLD})_\Gamma = (\mathsf{RLD})_{\mathrm{in}}.

Previously I proved an example demonstrating that (\mathsf{RLD})_\Gamma \ne (\mathsf{RLD})_{\mathrm{in}}, but this example is believed by me to be wrong. The example was removed from the book.

Thus I removed all references to (\mathsf{RLD})_\Gamma (as it is the same as (\mathsf{RLD})_{\mathrm{in}}) and reworked the chapter “Funcoids and filters” to reflect the change.

The book is available free of change at this Web page.

The story of the past:

(\mathsf{RLD})_\Gamma was defined by the formula (\mathsf{RLD})_\Gamma f = \bigsqcap^{\mathsf{RLD}} \mathrm{up}^\Gamma\, f.

From the theorem in “The diagram” section (the theorem with a diagram) it trivially follows that (\mathsf{RLD})_\Gamma f = (\mathsf{RLD})_{\mathrm{in}} f. It follows trivially, but I have found this only today.

September 10, 2016 / porton

An error in my book

I proved both (\mathsf{RLD})_\Gamma \ne (\mathsf{RLD})_{\mathrm{in}} and (\mathsf{RLD})_\Gamma = (\mathsf{RLD})_{\mathrm{in}}.

So there is an error in my math research book.

I will post the details of the resolution as soon as I will locate and correct the error. While the error is not yet corrected I have added a red font note in my book.

September 4, 2016 / porton

I’ve solved a conjecture about pseudodifference of filters

I claimed earlier that I partially solved this open problem.

Today I solved it completely. The proof is available in this PDF file.