
Re: Principal Reliods
Posted:
Nov 3, 2013 10:37 PM


On Sun, 3 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. Well ok, now that you've let us know your considing an ordered, proper subset of reloids, it makes sense.
> 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. >

