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Topic: Order, Filters and Reloids
Replies: 11   Last Post: Sep 1, 2013 10:54 AM

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Victor Porton

Posts: 621
Registered: 8/1/05
Re: Order, Filters and Reloids
Posted: Aug 29, 2013 6:08 AM
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William Elliot wrote:

> On Tue, 27 Aug 2013, Victor Porton wrote:
>> William Elliot wrote:
>>

>> >> > The composition of reloids is associative.
>> >> Correct. Theorem 7.13 in my book:
>> >> http://www.mathematics21.org/algebraic-general-topology.html

>> >
>> > How many theorems of chapter 7 are about reloids only?
>> > Other than what is mentioned here, what are some of them?

>>
>> Have you noticed that chapter 7 is titled "Reloids"? All theorems in this
>> chapter are about reloids.

>
> I'm sill hassling with the weirdly written section "Funcoids and Reloids".
> PDF
> files are a pain in the butt, hard to work with. I've yet to get anything
> out of it except noticing weird unneeded notations and definitions that
> make it all the more complicated.
>
> For example, what's (F,X,Y)^-1? Wouldn't it simply be (F^-1,Y,X)?
> Your definition of (F,X,Y)^-1 was unduly confusing and muddled.


(F,X,Y)^-1 is useful when we assign f=(F,X,Y). This way we have a definition
for the morphism f^-1.

> In addition, there's no efficient way to write and
> comment about interesting parts.
>
> If F,G in Ft(X), H in Ft(Y), it's straight forward to prove
> F/\G xx H = FxxH /\ GxxH.


Well, if you do NOT use reverse order which I recommend.

> Without referring to your pdf file, how do you prove:
> if, for all j, Fj in Ft(X), G in Ft(y), then
> (/\_j Fj) xx G = /\_j (Fj xx G)?


It becomes a trivial set-theoretic equation if we replace /\ -> \/

(\/_j Fj) xx G = \/_j (Fj xx G)?

It seems for me that you have confused a poset with the dual poset. However
I am not sure what you do mean.

> The same method used for the finite case, can't be reworked for the
> infinite
> case. BTW, the proposition is false if the index set for j is empty.


--
Victor Porton - http://portonvictor.org



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