
Re: Principal Reliods
Posted:
Nov 2, 2013 9:49 PM


On Sat, 2 Nov 2013, Victor Porton wrote: > William Elliot wrote:
> >> > 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. > >> > >> Right. > >> > >> > 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. > > > >> Yes, but if we limit our consideration to principal filters **only**, > >> then by definition any suprema and infima would be also principal. > > So you require that only infinums that are principal reloids to be > > accepted? That is not wise for, as shown above, principal reloids would > > not be closed under infinite infinums. Thus principal reloids aren't a > > complete lattice. > > You wanted to make a quantale out of principal reloids. To make it one need > to restrict suprema and infima only to principal reloids. The resulting > quantale is isomorphic to the quantale of binary relation, so it is > effectively nothing new. > It's not possible because infinite infinums of principal reloids isn't a pricipal reloid. In addition, for closure of compositions the reloids cannot be a filter on a product of different sets; they need to be a filter on the product of the same set.
> Topic closed.
There is an isomophism but it not a complete isomorphism.
Topic over.

