
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

