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




Re: Order, Filters and Reloids
Posted:
Aug 26, 2013 8:14 AM


William Elliot wrote:
> F is a filter for S when F subset P(S)\{empty set}, F not empty > and F ordered by subset is a down directed, upper set.
The improper filter (which I also call a filter) is P(S) that is it contains empty set.
> Since a not empty intersection of filters is again a filter, the > set of filters ordered by subset is a complete lower semilattice. > > If the empty set is allowed to be in filters, then there's a maximum > filter P(S), and the (partially) ordered set of filters is a complete > order. In what follows, "filter" will mean a filter in the usual sense > or the improper maximum filter P(S).
OK.
> Let F be a filter for X, G a filter for Y. > Then B = { UxV  U in F, G in Y } subset P(XxY) is a filter base > and the product of F and G, FxxG, is the filter generated by B. > > (F,X,Y) is a reloid when F is a filter for XxY. > > (F,X,Y) is a complete reloid when there's some A for which > A subset { GxxH  G principle ultrafilter for X, H ultrafilter for Y } > and F = /\A, the great intersecion of A, the infinum of A.
principle > principal
> The composition of two reloids (F,X,Y) and (G,Y,Z) > is the reloid (F,X,Y) o (G,Y,Z) = (H,X,Z) > where H is the filter generated by > { AoB  A in F, B in G } > and the compositon of A and B, > AoB = { (x,z) in XxZ  some y in Y with (x,y) in A, (y,z) in B }
Correct.
> Propositions. Are the following true? Are there proofs? > The product of filters is associative.
No, it is associative up to an isomorphism.
I have no formal proof for this, yet.
> The composition of reloids is associative.
Correct. Theorem 7.13 in my book: http://www.mathematics21.org/algebraicgeneraltopology.html
> If A is a set of filters for X, B a set of filters for Y, then > /\{ FxxG  F in A, G in B } = /\A xx /\B.
Yes, theorem 7.22 in my book.
> The composition of two complete reloids is complete.
It is a conjecture.
 Victor Porton  http://portonvictor.org



