"associativity in MMon" means then that every domain D that
implements MMon must be such that the relation EqUaL((a*b)*c,
a*(b*c)) holds for every elements a,b,c in D.
You know, we have something similar in aldor-combinat. Just suppose
that D is the domain containing all species. Ufff, no, we couldn't
claim MMon, because our implementation of * is neither commutative
nor associative.
Why not? The question is: commutative with respect to which notion
of equality.
Well, you may be right. But unless I see a mathematical definition of
such an EqUaL, I still cannot claim commutativity, right?
Of course, equality as implemented violates commutativity. (actually,
I think that's my whole point: it's not the implementation of
multiplication that violates commutativity, but rather the
implementation of equality.)
Ehm, I would somehow want that for domains where = is computabled then
EqUaL is the same relation as =.
... and on uniqueness of the unit 1? Can you claim that for
PowerSeries?
No. and there should be another assertion for that.
So, it's not a monoid?
Then for sure you can have this. And if you impose an explicit
order on the summands and multiplicants, you could define
"determinant" even for structures where neither * nor + is
commutative. To put it to the extreme, you can define such a
determinant expression even if neither * nor + is associative.
But what can you then do with this definition of determinant?
Again: not the implementation of * and + is bad. It's only equality
that's failing. If we do not use equality, all is well.
But equality is fundamental. How whould you define the concept of a set
if you don't have equality on the elements? You must have equality
somewhere. And I'd say, it better should be computable.
However, we have to rely on commutativity.
Why?
I cannot think of an example where it matters right now, but I'm
sure there are. (I should add: yes, there is an established
definition for determinants in noncommutative rings. But I never
worked with such beasts so far.)
That was not my point. I could also define
determinant(x: %): R == 1
But that is probably not a very useful concept of "determinant". But
it's a perfect definition or rather implementation of a determinant
function.
In any case, the current implementations of the determinant almost
certainly only give the right result in the commutative case.
What is "right"?
Ralf
PS: Let's stop here. It doesn't lead to anything.
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