
Re: Principal Reliods
Posted:
Nov 1, 2013 12:00 AM


On Thu, 31 Oct 2013, Victor Porton wrote: > William Elliot wrote: > > > On Wed, 30 Oct 2013, Victor Porton wrote: > >> William Elliot wrote: > > > >> >> It is by definition of principal reloids. A principal reloids is by > >> >> definition a reloid corresponding to a binary relation. Trivially this > >> >> correspondence is bijective. > > > > It seems that principal reloids bijective map to principal relations. > What is principal relation? > A relation of the form AxB.
> Principal reloids bijective map to all relations. > > >> > Is this corect? > >> > F is ia principal reloid for XxY when F is the principal filter of AxB > >> > for some A subset X, B subset Y. > >> > >> No, no need to introduce X and Y. > > Why? > > > >> F is is a principal reloid for AxB when F is a principal filter for AxB. > > Strictly speaking, a reloid is (A;B;F) where F is a principal filter on AxB. > No X, Y here.
I'm using X and Y in place of A and B because of the established usage of X and Y to be the space in discussion which leaves A and B free to use as subsets.
> > Don't understand. By Definition 7.5 I thought that a reloid for XxY > > was principal when the reliod was F_AxB for some A and B. > > Principal reloid is a reloid induced by some binary relation. That binary > relation is not necessarily a product.
> > Would you describe what Definition 7.5 means in terms that I can > > understand? > > Again: Principal reloid is a reloid induced by some binary relation.
Ok we agree. You notation is terribly complicated and were you to use less of it and ocassionally more words, your text would be much easier to read and more likely read.
> >> >> So we have kinda isomorphism between binary relations and principal > >> >> reloids. > >> > > >> > For composition and finite intersections but not infinite > >> > intersections. > >> > >> No. If we limit to the poset of principal filters, it is also lattice > >> isomorphism. > > > > But not a complete lattice isomorphism. > > It is a complete lattice isomorphism (if we consider lattice operations on > principal filters only).
F_r = F((r,r)) is a principal filter for R.
The filter . . /\_(0<r) F_r = { (a,b)  a < 0 < b } } is not principal. Casing this into reloids, . . /\_(0<r) ({R} xx F_r) = {R}/\_(0<=r) F_r
is the infinum of principal reloids that's not a principal reloid.

