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Topic: About generalizations
Replies: 21   Last Post: Aug 20, 2013 11:45 PM

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William Elliot

Posts: 1,705
Registered: 1/8/12
Re: About generalizations
Posted: Aug 20, 2013 11:45 PM
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On Tue, 20 Aug 2013, Victor Porton wrote:
> William Elliot wrote:
> > On Mon, 19 Aug 2013, William Elliot wrote:
> >> On Mon, 19 Aug 2013, Victor Porton wrote:
> >> > >> >
> >> > >> >> A complete reloid is a join (on a complete lattice of reloids
> >> > >> >> between two fixed sets) of (reloidal) products of a trivial
> >> > >> >> ultrafilter and a (non necessarily trivial) ultrafilter.

> >>
> >> For Ft(XxY) = { F | F filter for XxY } to be a complete order by
> >> inclusion, doesn't Ft(XxY) have to include both P(XxY) and the empty set?

> >
> > No, the empty filter isn't needed for the bottom of Ft(XxY) is {P(XxY)}
> > and the top is P(XxY).

>
> No, the bottom of Ft(XxY) is {XxY} and the top is P(XxY).


Correct.

> > However to define a complete reloid, P(X,Y) in Ft(XxY) isn't needed.
> > Indeed, Ft(S) with P(S) excluded and subset order is a complete, down or
> > lower, semi-lattice because the intersection of any number of filters
> > is again a filter, that is intersection is the meet.

>
> I don't understand anything in the above paragraph.


In awhile, I'll start a new thread "Reloids" summarizing what I've learned
about reloids and filters as applied to reloids with some propositions,
speculations and perhaps some proofs.

> > BTW, /\{ F | F principal ultrafilter for S } = {P(S)}
> > that is, the meet of all principal ultrafilters is the trivial
> > filter containing but one subset.

>
> = {S} not {P(S)}.


Correct again.



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