Re: P1788: Our first formal motion: IR versus *IR

[Ulrich Kulisch <Ulrich.Kulisch@xxxxxxxxxxx>]

Dear all,

I think we are talking about mathematics. If mathematics is viewed as 
the science of the structures then IR as it is defined in the 
StandardNotation is an  incomplete entity. With respect to set inclusion 
as an order relation it is an ordered set. But the empty set, for 
instance, which may be the result of an intersection is excluded.

A more natural definition of IR would be to consider it as a complete 
lattice with set inclusion as the order relation. Then the least element 
is the empty set and the greatest element is the set R = (-oo, +oo).
The infimum in IR is the intersection and the supremum is the interval 
(convex) hull. The intersection may lead to the empty set which is an 
element of IR. IR is a subset of the power set PR which is also a 
complete lattice with respect to set inclusion as the order relation. 
The infimum in PR and in IR is always the same.

For my view of the matter see the attached brief article.

Best whishes
Ulrich Kulisch


Nate Hayes schrieb:
> > The suggestion is:
> > 1. to use the notation IR for the complete lattice of closed (bounded 
> > and unbounded) intervals over R with the empty set as the least 
> > element and the set  R = (-oo, +oo) as the greatest element.
> > 2. to allow an index notation for intervals as an alternative.
> >
> > As far as I understand an earlier mail by Arnold Neumaier;
> > the proposed StandardNotation uses the notation *IR for the set 
> > defined under 1. It restricts the symbol IR to the subset of non 
> > empty, closed and bounded intervals over R.
>
> I now see more clearly the subtle distictions people are trying to make.
>
> On the one hand, I do agree with Arnold Neumaier that the traditional 
> usage of IR is the set of non empty, closed and bounded intervals over 
> R. This also conincides with the traditional definition of KR, which 
> is the set of non empty, closed and bounded Kaucher intervals over R. 
> Note that in KR intersection is already closed with respect to the 
> inclusion relation.
>
> On the other hand, open endpoints are useful for unbounded intervals, 
> e.g., (-oo,a] or [a,+oo) etc. This is true for both classical and 
> Kaucher intervals. However, only the classical intervals also require 
> the empty set.
>
> One more thing to consider is that Intervals with non-infinite open 
> endpoints are generally unnescessary in both IR and KR.
>
> However, I don't see that the standard notation document specifies if 
> infinities are actually members of the interval in *IR or not. It 
> almost seems to me the document specifies *IR as the union of IR with 
> the empty and extended-real intervals. The extended-real intervals are 
> different than the unbounded intervals, because the former contain 
> infinity as a member and the latter do not. The former also do not 
> include the set of open intervals, but the latter do. For these 
> reasons, it seems there still is confusion about what the standard 
> meaning of *IR is (or at least the definition provided in the standard 
> notation document could be a little more explicit about what it means).
>
> So I guess I'm happy to stick with traditional meanings for IR and KR, 
> but then to be flexible in notation of *IR as long as the meaning of 
> it is made explicit. Also, it seems that *KR would be the natural 
> notation for the KR analogy.
>
> Sincerely,
>
> Nate Hayes

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