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Topic: Please nominate me for Abel Prize
Replies: 33   Last Post: Jul 19, 2013 10:22 PM

 Messages: [ Previous | Next ]
 Victor Porton Posts: 621 Registered: 8/1/05
Re: Please nominate me for Abel Prize
Posted: Jul 18, 2013 7:56 AM

William Elliot wrote:

> On Wed, 17 Jul 2013, Victor Porton wrote:
>> William Elliot wrote:
>> >
>> >> Reloid is basically just a filter on a Cartesian product of two sets.
>>
>> The exact definition:
>>
>> A reloid is a triple (A;B;F) where A and B are sets and F is a filter on
>> their product AxB.
>>
>> (Note that I allow a filter to be improper.)

>
> What's an improper filter?

Improper filter on a set A is the set A itself.

> (A,B,F) is a reloid when F is a filter for AxB or an improper filter for
> AxB.
>

>> A funcoid is a quadruple \$(A;B;a;b)\$ where \$A\$ and \$B\$ are sets, \$a\$ is a
>> function from the set of filters on \$A\$ to the set of filters on \$B\$,
>> \$b\$
>> is a function from the set of filters on \$B\$ to the set of filters on
>> \$A\$, subject to the following condition:

>
> Skip the TeX is these ascii only newsgroup; it make reading harder.
>
> Let FF(A) = { F | F filter over A }. Are improper filters included?

Yes, in all my works improper filter is included.

By the way, it was somebody's stupid idea to exclude improper filter from
the set of filters.

>> For every filter X on A and every filter Y on B, the filter (on B)
>> generated by the union of Y and a(X) is not the entire power set P(B) iff
>> the filter (on A) generated by the union of X and b(Y) is not the entire
>> power set P(A).

>
> If S is a set, B subset P(S), then
> filter< B > is the filter for S generated by B.
>
> (A,B,f,g) is a funcoid when f:FF(A) -> FF(B), g:FF(B) -> FF(A),
> for all X in FF(A), Y in FF(B),
> filter< Y \/ f(X) > proper subset P(B)
> iff fllter< X \/ g(Y) proper subset P(A)

Yes.

> (A,B,f,g) is a funcoid when f:FF(A) -> FF(B), g:FF(B) -> FF(A),
> for all X in FF(A), Y in FF(B),
> filter< Y \/ f(X) > = P(B) iff fllter< X \/ g(Y) > = P(A).
> Is that correct?

Yes.

Two definitions which you presented are equivalent to each other.

>> See http://nlab.mathforge.org/nlab/show/funcoid for a more understandable
>> definition of funcoids than one in the paragraph above of this post.

>
> No thanks, I've not the patience to mess with pdf stuff.

The above link is HTML with MathML, not PDF.

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