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Topic: Which term to choose?
Replies: 41   Last Post: Nov 9, 2013 5:20 AM

 Messages: [ Previous | Next ]
 William Elliot Posts: 2,637 Registered: 1/8/12
Re: Principal Reliods
Posted: Nov 3, 2013 10:37 PM

On Sun, 3 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.

>
> Suprema and infima depends on the poset on which they are taken.
>
> If we take suprema and infima on the poset of principal reloids, then the
> suprema and infima are by definition principal reloids.

Well ok, now that you've let us know your considing
an ordered, proper subset of reloids, it makes sense.

> This is obviously a quantale bijective to the quantale of binary relations.
>

> >> Topic closed.
> >
> > There is an isomophism but it not a complete isomorphism.
> >
> > Topic over.

>
>

Date Subject Author
10/25/13 Victor Porton
10/25/13 Peter Percival
10/25/13 fom
10/25/13 William Elliot
10/26/13 William Elliot
10/26/13 Victor Porton
10/26/13 William Elliot
10/27/13 Victor Porton
10/27/13 William Elliot
10/28/13 Victor Porton
10/29/13 William Elliot
10/29/13 Victor Porton
10/30/13 William Elliot
10/30/13 Victor Porton
10/30/13 William Elliot
10/31/13 Victor Porton
11/1/13 William Elliot
11/1/13 Victor Porton
11/1/13 William Elliot
11/2/13 Victor Porton
11/2/13 William Elliot
11/3/13 Victor Porton
11/3/13 Victor Porton
11/3/13 William Elliot
11/4/13 William Elliot
11/4/13 Victor Porton
11/5/13 William Elliot
11/5/13 Victor Porton
11/6/13 William Elliot
11/6/13 Victor Porton
11/6/13 William Elliot
11/7/13 Victor Porton
11/7/13 William Elliot
11/8/13 William Elliot
11/8/13 Victor Porton
11/8/13 William Elliot
11/9/13 Victor Porton
11/9/13 William Elliot
11/9/13 William Elliot
11/9/13 Victor Porton
10/26/13 Victor Porton