
Re: Principal Reliods
Posted:
Nov 2, 2013 8:10 AM


William Elliot wrote:
> On Fri, 1 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 > > (Correction made.) > >> > 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.
Topic closed.

