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

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William Elliot

Posts: 1,606
Registered: 1/8/12
Re: Principal Reliods
Posted: Nov 1, 2013 12:00 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

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.



Date Subject Author
10/25/13
Read Which term to choose?
Victor Porton
10/25/13
Read Re: Which term to choose?
Peter Percival
10/25/13
Read Re: Which term to choose?
fom
10/25/13
Read Re: Which term to choose?
William Elliot
10/26/13
Read Re: Which term to choose?
William Elliot
10/26/13
Read Re: Which term to choose?
Victor Porton
10/26/13
Read Re: Which term to choose?
William Elliot
10/27/13
Read Re: Which term to choose?
Victor Porton
10/27/13
Read Re: Which term to choose?
William Elliot
10/28/13
Read Re: Which term to choose?
Victor Porton
10/29/13
Read Re: Which term to choose?
William Elliot
10/29/13
Read Re: Which term to choose?
Victor Porton
10/30/13
Read Principal Reliods
William Elliot
10/30/13
Read Re: Principal Reliods
Victor Porton
10/30/13
Read Re: Principal Reliods
William Elliot
10/31/13
Read Re: Principal Reliods
Victor Porton
11/1/13
Read Re: Principal Reliods
William Elliot
11/1/13
Read Re: Principal Reliods
Victor Porton
11/1/13
Read Re: Principal Reliods
William Elliot
11/2/13
Read Re: Principal Reliods
Victor Porton
11/2/13
Read Re: Principal Reliods
William Elliot
11/3/13
Read Re: Principal Reliods
Victor Porton
11/3/13
Read Re: Principal Reliods
Victor Porton
11/3/13
Read Re: Principal Reliods
William Elliot
11/4/13
Read Principal Reliods
William Elliot
11/4/13
Read Re: Principal Reliods
Victor Porton
11/5/13
Read Re: Principal Reliods
William Elliot
11/5/13
Read Re: Principal Reliods
Victor Porton
11/6/13
Read Partition of a filter
William Elliot
11/6/13
Read Re: Partition of a filter
Victor Porton
11/6/13
Read Re: Partition of a filter
William Elliot
11/7/13
Read Re: Partition of a filter
Victor Porton
11/7/13
Read Re: Partition of a filter
William Elliot
11/8/13
Read Partition of a filter
William Elliot
11/8/13
Read Re: Partition of a filter
Victor Porton
11/8/13
Read Re: Partition of a filter
William Elliot
11/9/13
Read Re: Partition of a filter
Victor Porton
11/9/13
Read Re: Partition of a filter
William Elliot
11/9/13
Read Re: Partition of a filter
William Elliot
11/9/13
Read Re: Partition of a filter
Victor Porton
10/26/13
Read Re: Which term to choose?
Victor Porton
11/4/13
Read Re: Which term to choose?
aliahmadikram

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