Date: Oct 30, 2013 7:24 AM
Author: Victor Porton
Subject: Re: Principal Reliods

William Elliot wrote:

> On Tue, 29 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.

>
> 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.

F is is a principal reloid for AxB when F is a principal filter for AxB.

> 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.

>> 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.

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.

>> 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.