
Re: About generalizations
Posted:
Aug 20, 2013 11:45 PM


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, semilattice 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.

