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

 Messages: [ Previous | Next ]
 Victor Porton Posts: 621 Registered: 8/1/05
Re: Principal Reliods
Posted: Oct 31, 2013 8:52 AM

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?

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.

> 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.

>> > 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.

Filters on cartesian products and reloids are essentially the same.

>> >> 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 a complete lattice isomorphism (if we consider lattice operations on
principal filters only).

>> 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.

I don't understand. What abstract algebra?

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