
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.

