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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: Nov 1, 2013 11:04 AM

William Elliot wrote:

> 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.

Why do you call it principal relation?

I recall that a principal reloid is a reloid corresponding to an
**arbitrary** binary 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.

>> 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.

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.

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