Hi Nicolas,

Thank you for your reply; it was very useful.

My main problem now is that I don't know how to implement a new parent.
Specifically a new monoid. I have looked at FreeMonoid and all I really got 
out of this was how to call the function.

I am still unclear about parents and categories. As I understand it these 
must be implemented using classes
(including abstract classes and/or metaclasses which are still classes).

Looking in FreeMonoid I find FreeMonoid_class and FreeMonoidFactory. I don't 
know what a factory is so I have to guess.
My guess is that it is a function that returns a class. This would then give 
a construction of classes depending on parameters
(for FreeMonoid the parameters are the set of generators).

To construct a single monoid I would expect to define a class whose parent 
is Monoid and then to write the class methods
(in this case one, returning the unit and mul, returning the product).
There are several things Monoid or Monoids and I don't know what their roles 
are.

However I would really want to construct monoids depending on parameters 
(which I would have thought was typical)
and I don't know where to start with this.

I will reply on sage-support to your answer as you requested.

I am amazed that posets is immutable. How is that done?
I have not yet discovered how labelled posets are constructed.
Have I missed something?

Thank you for your patience with me.

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