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

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William Elliot

Posts: 2,637
Registered: 1/8/12
Re: About generalizations
Posted: Aug 20, 2013 5:15 AM
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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).

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.

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.

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