[Aldor-l] Aldor-Meet

[Ralf Hemmecke]

Hi Oleg,

On 11/20/2007 05:15 PM, Oleg Golubitsky wrote:
> The two definitions of `Meet(C1,C2)':
> 1) a category which has the *intersection* of exports of `C1' and `C2'
> 2) a category `C' such that
>     D has C <=>  D has C1  or  D has C2
>     (in other words: C has *all exports of C1* or *all exports of C2*)
> are actually not equivalent: "2)" implies "1)" but not vice versa.

Well observed.

Now with definition 1) and

define C: Category == Meet(C1,C2);
D1: C1 == add {...}

should
   a) "D1 has C"            give true or false,
   b) "D1 has Meet(C1,C2)"  give true or false?

> The above test has shown that the compiler does not require "2)".
> However, it might make sense to do so, for two reasons:
> i. if we use definition "1)" then it might be hard to find
>    good examples for `Meet', because one can always
>    define category `C' first, and then define `C1' and `C2'
>    both as `C with { something }' instead of defining first
>    `C1' and `C2' and using `C == Meet(C1,C2)'.

That is simply wrong. I'll write a library abc, and give you the file 
libabc.al. In abc I define the categories
Group, SemiGroup, and RingWithOne and domains D1:Cat1,..., Dn: Catn 
where each of Cati exports

SemiGroup with {1:%}

See
http://aldor.org/pipermail/aldor-l_aldor.org/2007-November/000817.html

Now you would like to define "Monoid". How can you do it *before*?

Clearly, you could use "extend" for D1 up to Dn. But you would not like 
to do this if n is relatively big or changes with the next release of my 
library.

> ii. for definition "2)" there is an example.
>     A bivariate polynomial can be viewed as an element of
>      a) R[x,y]    (free module with the monomial basis)
>      b) R[x][y]   (polynomial ring over the univariate polynomial ring)
>     So one could define the category of bivariate polynomials
>     as the `Meet' of "a)" and "b)".

That doesn't sound too convincing. I rather have an inefficient
   deg: % -> N^n
function than non at all.

Ralf