Implement Algebra of Words and some linear operators

[bc9eee3a-171b-423e-8ec5-1c11eaa42e98]

Implement the algebra R for a ring R and an alphabet A. We also implement linear operators e, f, sigma and zeta as defined in [Lothaire, M. (2011), Algebraic combinatorics on words, Encyclopedia of Mathematics and Its Applications 90, Cambridge University Press].

CC: @seblabbe @anneschilling @nthiery @videlec @tscrim

Component: combinatorics

Author: virmaux

Branch/Commit: u/virmaux/16179 @ 5e4168c

Issue created by migration from https://trac.sagemath.org/ticket/16179

[bc9eee3a-171b-423e-8ec5-1c11eaa42e98]

New commits:

aea1c8f	beginning of work on words_algebra
fc66439	refactorization in a category, added operators e, f, sigma, zeta on words algebra
5e4168c	documentation

[bc9eee3a-171b-423e-8ec5-1c11eaa42e98]

Branch: u/virmaux/16179

[bc9eee3a-171b-423e-8ec5-1c11eaa42e98]

Commit: 5e4168c

[videlec]

comment:4

Hi Aladin,

You did good about the tests in the documentation! As I am (as well as many others) not familiar with algebras it would be nice to explain what are the e, f, zeta and sigma operators in the docstring of the methods. (I like learning math by digging in Sage documentation but hate having to guess what are the specifications of a function)

For the design, most piece of code related to categories (which intend to be generic code) must go inside sage/categories or for some rare case in sage/structure. You decided to implement the category of your object as a subclass of itself... which means that the code is not generic. So I would not put that as category code but directly inside the Element. If you have a strong argument about creating a category for the FreeAlgebra please explain it on both this ticket and the file where the code belongs to. On the other hand, in his axioms related peregrinations in #10963, Nicolas proposed to have a .Free() (meaning that I would be able to do Groups().Free() to choose my category). So your code might move to somewhere more appropriate in the future.

There already is a FreeAlgebra in sage.algebras.free_algebra. What I would naturally do is to move your ElementMethods as methods of FreeAlgebraElement in sage.algebras.free_algebra_element. If you are not happy with the current FreeAlgebra which is rather old, it would be nice to refresh it a bit.

Vincent

[tscrim]

comment:6

FYI #14848 makes FreeAlgebra into a subclass of CombinatorialFreeModule.

I do have a question. Are you considering infinite words as elements of this algebra? If not, then I think you should add methods/functionality directly to FreeAlgebra (where the arg names gives the alphabet).

[bc9eee3a-171b-423e-8ec5-1c11eaa42e98]

comment:7

Thank you for the comments, I will play with FreeAlgebra and will try to use it.