
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?

