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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: Oct 30, 2013 11:46 PM

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.

> > 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.

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.

Would you describe what Definition 7.5 means in terms that I can understand?

> > Though all principal reloids are principal filters, there are some
> > principal filters that aren't principal reloids. Have you a example?

>
> Yes, filter {{0}} is not a reloid.
>

It's not a filter for a product.

> >> This correspondence also maps composition of binary relations to
> >> composition of reloids (easy to show).

> > There's also a bijection between principle filters for a product
> > and binary relations of the product. Is this also isomorphic like?

>
> Yes.
>

> >> 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 not an isomorphism in the case when we consider not only principal
> reloids.
>

> > Are there any theorems true for principal reloid for a product
> > that fail for pricnipal filters for the product?

>
> Not sure that I understand you.
>

> >> > It seems there's enough material about reloids for a short paper.
> >>
> >> Seriously, why to write such a paper when there is already a whole book?

> >
> > The book is too long and complex to get the attention of many people.
> > It also needs to be recased into existing notation and terminalogy.
> >

> >> Moreover reloids are closely related with funcoids, and to consider
> >> reloids without funcoids would be wrong.

> >
> > Isn't there some use of reloids per se?

>
> Not sure.

It would be good if there were for then the topic of reloids would
be more than an abstract algebra.

> >> In fact such a paper already exists (and it is in peer review now):
> >> http://www.mathematics21.org/algebraic-general-topology.html
> >> "Funcoids and Reloids" (PDF, preprint)

> >
> > Is this just the chapters on funcoids and reloids of your book
> > or is it much different?

>
> It is an older version of some chapters from my book. Also in the article
> the notation is different.

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