
Re: Please nominate me for Abel Prize
Posted:
Jul 17, 2013 11:50 PM


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?
(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?
> 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)
(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?
> 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.

