pickling extension classes

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Even when you had a Python class,

loads(dumps(a))==a

So, if one sees your problem as a Python/Cython problem, then there are two parts of your problem (which are mostly answered by the documentation of Python, by the way). If you work in Sage, some related tools are available, so that your problem has a third part.

The first part is comparison of objects. For that purpose, Python (and Cython) use methods such as __cmp__ and __richcmp__ (see Python docs). If they are not provided, comparison will simply compare id(...).

The second part is pickling/serialisation. Python offers different serialisation protocols (again: see Python docs).

If you have Python classes, you may use the first protocol, which relies on methods like __getstate__, __setstate__, __getinitargs__, __getnewargs__. But for extension classes you have to use the second protocol: You need to provide a __reduce__ method, returning a function f and arguments args to that function, so that f(*args) will return (a copy of) the original object.

Here is the third part of your problem: You work in Sage. You know that they say "we don't re-invent the wheel, we build the car". Thus, it makes sense to use Sage's infrastructure, rather than programming everything from scratch.

For example, when you want to create a class whose instances are "unique" (meaning: A loads(dumps(...)) copy of the instance will not only be equal but identical with the instance), you can simply inherit from sage.structure.unique_representation.UniqueRepresentation. In more complicated cases (or with Cython classes), you can also use sage.structure.factory.UniqueFactory, which does not seem to appear in the reference manual, but you can of course still browse its documentation in a Sage session.

If your intension is to implement algebraic structures, you are likely to derive your classes from sage.structure.element.Element or sage.structure.parent.Parent (or from existing sub-classes; also see this worksheet). In that case, you should look into the source code, near to the __cmp__ and __richcmp__ methods: You will find comments that explain what methods you should provide for comparison.