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December 1, 2016 / porton

An informal open problem in mathematics

Characterize the set \{f\in\mathsf{FCD} \mid (\mathsf{RLD})_{\mathrm{in}} f=(\mathsf{RLD})_{\mathrm{out}} f\}. (This seems a difficult problem.)

December 1, 2016 / porton

A new theorem proved

I have proved (\mathsf{RLD})_{\mathrm{in}} \Omega^{\mathsf{FCD}} = \Omega^{\mathsf{RLD}} (where \Omega^{\mathsf{FCD}} is a cofinite funcoid and \Omega^{\mathsf{RLD}} is a cofinite reloid that is reloid defined by a cofinite filter).

The proof is currently available in this draft.

Note that in the previous draft there was a wrong formula for (\mathsf{RLD})_{\mathrm{in}} \Omega^{\mathsf{FCD}}.

December 1, 2016 / porton

A typo in my math book

I’ve found a typo in my math book.

I confused existential quantifiers with universal quantifiers in the section “Second product. Oblique product” in the chapter “Counter-examples about funcoids and reloids”.

December 1, 2016 / porton

More on generalized cofinite filters

I added more properties of cofinite funcoids to this draft.

November 30, 2016 / porton

Generalized cofinite filters

I have described generalized cofinite filters (including the “cofinite funcoid”).

See the draft at http://www.mathematics21.org/binaries/addons.pdf

November 26, 2016 / porton

A new diagram about funcoids and reloids

Define for posets with order \sqsubseteq:

  1. \Phi_{\ast} f = \lambda b \in \mathfrak{B}: \bigsqcup \{ x \in \mathfrak{A} \mid f x \sqsubseteq b \};
  2. \Phi^{\ast} f = \lambda b \in \mathfrak{A}: \bigsqcap \{ x \in \mathfrak{B} \mid f x \sqsupseteq b \}.

Note that the above is a generalization of monotone Galois connections (with \max and \min replaced with suprema and infima).

Then we get the following diagram (see this PDF file for a proof):

diagram2

It is yet unknown what will happen if we keep apply \Phi_{\ast} and/or \Phi^{\ast} to the node “other”. Will this lead to a finite or infinite set?

November 4, 2016 / porton

A new kind of product of funcoids

The following is one of a few (possibly non-equivalent) definitions of products of funcoids:

Definition Let f be an indexed family of funcoids. Let \mathcal{F} be a filter on \mathrm{dom}\, f. a \mathrel{\left[ \prod^{[\mathcal{F}]} f \right]} b \Leftrightarrow \exists N \in \mathcal{F} \forall i \in N : \mathrm{Pr}^{\mathsf{RLD}}_i\, a \mathrel{[f_i]} \mathrm{Pr}^{\mathsf{RLD}}_i\, b.
for atomic reloids a and b.

Today I have proved that this really defines a funcoid. Currently the proof is present in draft of the second volume of my book,

A probably especially interesting case is if \mathcal{F} is the cofinite filter. In this way we get something similar to Tychonoff product of topological spaces.

This may possibly have some use in study of compact funcoids.