[Axiom-math] Re: [fricas-devel] Re: [open-axiom-devel] [fricas-devel] Re: [fricas-devel] Re: iterators and cartesian product.

[Bill Page]

On 10/22/07, Ralf Hemmecke wrote:
>
> On 10/22/2007 06:20 PM, Gabriel Dos Reis wrote:
> > On Mon, 22 Oct 2007, Bill Page wrote:
> >
> > |
> > | On 22 Oct 2007 10:16:33 -0500, wrote:
> > | > Bill Page writes:
> > | > ...
> > | > |
> > | > | I would like to consider what is?
> > | > |
> > | > |   1..9
> > | > |
> > | > | Right now in Axiom this is evaluated as a member of 'Segment
> > | > | PositiveInteger', i.e. the domain of all such segments. But in general
> > | > | I think I would prefer if '1..9' actually denoted a domain - a subset
> > | > | of the Positive Integers - with members 1, 2, 3 ... etc.
>
> Bill, you don't want 1..9 to be a domain. Of course, you can have it if
> you really want, but that just sounds like a domain of the set (or list)
> of the first 9 numbers. What would be the exports of this domain?
>

Yes, under this proposal '1..9' would be a domain consisting of the
set of the first 9 positive integers. I want to consider the view that
this is the same as thinking of 'PositiveInteger' as a sub-domain of
'Integer'. So like 'PositiveInteger', '1..9' would inherit some of the
structure of Integer, specifically 'OrderedSet', plus exports from
'Finite' and the operations 'high', 'low' and 'incr' of
'SegmentCategory'. However 'SEGMENT', 'BY' and 'convert' of the
existing 'SegmentCategory' would have to become domain constructors.

In general I am wondering about "set-like" objects and whether these
should always be modeled as domains.

> > | > Couold you elaborate on why `1..9' should denote a domain, and what
> > | > the benefits would be?
> > | >
> > |
> > | Well, for one I could then write the cross-product of such domains:
> > |
> > |   Product(1..9,1..4)
>
> > What would be its meaning?
>

If '1..9' and '1..4' are domains then the meaning of 'Product' is
already given by the existing domain constructor 'Product' in the
Axiom library

> ...

Regards,
Bill Page.