
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/algebraicgeneraltopology.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.
Oh, then I've already seen it from your text.

