Date: Nov 3, 2013 10:50 AM
Author: Victor Porton
Subject: Re: Principal Reliods

William Elliot wrote:

> On Sat, 2 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.

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.


Well, there are nothing interesting about this.