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

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Victor Porton

Posts: 621
Registered: 8/1/05
Re: About generalizations
Posted: Aug 20, 2013 8:41 AM
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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).

> 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.

> 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)}.

Victor Porton -

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