I think, that is a good example to explain how I understand “canonical”. Order relations are mathematical objects, which and they interrelate with other mathematical objects, like additive structures. There is a theory behind that, which works like a mobile in equilibrium - if you change one little part, it gets in motion and a lot of things do not work any more. That is why I am reluctant to extend the domain of certain functions, which handle mathematical objects to types, which do not behave “mathematically”.
Let me come back to the original subject: how and if should
cmp be defined for
I would recommend, to keep the following contract valid:
For each a and b of the same type, which is in the domain of isless/cmp (ìsless(::T,::T) where T) exactly one of the following statements shall be true
isless(a,b) === true or
isless(b,a) === true or
isequal(a, b) === true.
cmp(a,b) < 0 <=> isless(a,b) and
cmp(a,b) > 0 <=> isless(b,a) and
cmp(a,b) == 0 <=> isequal(a,b).
Besides that, of course associatitvity, antisymmetry, reflectivity has to be respected.
This contract can be followed by a lexicographical order for arrays, if you make the following restrictions:
- Arrays of different shape cannot be compared (throw exception).
- the elements are iterated in a defined fashion, which must be clear to the user, and compared element-wise until the first difference is detected.
- The element type must follow the above contract about isless/cmp.
It is an open decision, if we restrict that to one-dimensional arrays (only one of the
size(a) elements is different form
Tuple, the decision to use lexicographical ordering has been made long ago and is ok. Tuples are in my reception less “mathematical” than arrays and vectors and serve different purposes.
For complex numbers, ordering would be surprising and on purpose. If there is a reason to present or process a set of complex numbers in a certain order, there are probably as many different orders as there are application uses. For example I once sorted eigenvalues first by abs of the real part, then by the negative of the imaginary part - and that was appropriate for my use case.
I would prefer to not provide a standard ordering, but delegate that to the user. Reason