Renaming classes

[Antonio-JP]

Description of the issue

After showing the code to collaborators, we agreed that the names in the software are a bit counter-intuitive. Since the main goal of the package is to be able to represent objects from difference, differential and difference-differential algebra, the name "RingWithOperator" seems a bit difficult to follow and can lead to a misunderstanding with a Ring of Operators.

Hence, we have decide to attach to the following notation:

Definition Let $R$ be a ring and $\Delta = {\sigma_1,\ldots,\sigma_n}$ be a set of additive homomorphism, i.e., $\sigma_i(a+b) = \sigma_i(a) + \sigma_i(b)$. We call the pair $(R, \Delta)$ a d-ring and we say $n$ is the number of operation in the d-ring.

Some examples of possible $\sigma$ include:

• Homomorphisms: when $\sigma(ab) = \sigma(a)\sigma(b)$.
• Derivations: when $\sigma(ab) = \sigma(a)b + a\sigma(b)$.
• Skew derivation: when there is an homomorphism $\tau$ such that $\sigma(ab) = \sigma(a)b + \tau(a)\sigma(b)$.

Definition Let $(R,\Delta)$ be a d-ring. We define the ring of d-polynomials in $u$ as the new d-ring $(R[\Delta(u)], \Delta)$, where $\Delta(u) = {\sigma^\alpha(u)\ :\ \alpha\in\mathbb{N}^n}$ and the operations in $\Delta$ act naturally on the new variables.

Goals of the issue

• Change the name of RingWithOperators, and similar classes in ring_w_operator.py to DRings and similar.
• Change the name of classes with name RWOPolynomial... to DPolynomial.
• Keep an alias for all the old classes
• Set the deprecation system from the deprecation Python module for the old classes.