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October 19, 2014 / porton

Todd Trimble’s commentary analyzed

I have written a short article with my response on Todd Trimble’s commentary on my book.

In this response I present these of Todd Trimble’s results which are new for me.

Note that I skipped specifically category-theoretic results (such as that the category of endofuncoids is topological). I am going to amend my article with categorical results later.

October 11, 2014 / porton

Todd Trimble’s commentary on my math research

Todd Trimble has notified me that he has written a “commentary” (notes) on my theory of funcoids presented in my monograph.

His commentary is available at this nLab wiki page.

I’ve started to read his notes. First I needed to lookup into Wikipedia to know what Chu space is. He uses category theory however as it now seems to me not very advanced. So I hope to understand his writing in soon time.

September 26, 2014 / porton

Funcoid corresponding to reloid through lattice Gamma

Conjecture For every funcoid f and filter \mathcal{X}\in\mathfrak{F}(\mathrm{Src}\,f), \mathcal{Y}\in\mathfrak{F}(\mathrm{Dst}\,f):

  1. \mathcal{X} \mathrel{[(\mathsf{FCD}) f]} \mathcal{Y}      \Leftrightarrow \forall F \in \mathrm{up}^{\Gamma (\mathrm{Src}\, f ; \mathrm{Dst}\,      f)} f : \mathcal{X} \mathrel{[F]} \mathcal{Y};
  2. \langle (\mathsf{FCD}) f \rangle \mathcal{X} = \bigsqcap_{F      \in \mathrm{up}^{\Gamma (\mathrm{Src}\, f ; \mathrm{Dst}\, f)} f} \langle F \rangle      \mathcal{X}.
September 26, 2014 / porton

Conjecture: Restricting a reloid to Gamma before converting it into a funcoid

Conjecture (\mathsf{FCD}) f = \bigsqcap^{\mathsf{FCD}} (\Gamma (A ; B)  \cap \mathrm{GR}\, f) for every reloid f \in \mathsf{RLD} (A ; B).

September 25, 2014 / porton

A new conjecture about relationships of funcoids and reloids

Conjecture (\mathsf{RLD})_{\mathrm{in}} f = \bigsqcap^{\mathsf{RLD}} \mathrm{up}^{\Gamma (\mathrm{Src}\, f ; \mathrm{Dst}\, f)}\, f for every funcoid f.

(I use notation from this note and this draft article.)

September 25, 2014 / porton

Funcoids as filters and composition

I have recently proved that there is an order isomorphism between funcoids and filters on the lattice of finite unions of Cartesian products of sets.

Today I’ve proved that this bijection preserves composition.

See this note (updated) for the proofs.

September 16, 2014 / porton

Two new theorems about complete reloids

I have just proven the following two new theorems:

Theorem Composition of complete reloids is complete.

Theorem (\mathsf{RLD})_{\mathrm{out}} g \circ (\mathsf{RLD})_{\mathrm{out}} f = (\mathsf{RLD})_{\mathrm{out}} (g \circ f) if f and g are both complete funcoids (or both co-complete).

See this note for the proofs.

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