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