
Re: About generalizations
Posted:
Aug 20, 2013 5:15 AM


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, semilattice 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.

