User:Hillgentleman/computer algebra

on using the computer to study abstract algebraic structures

google: google:algebraic+structure+data+type, google:object+oriented+algebraic+structure
Modelling Algebraic Structures. in a Symbolic Computation Environment by Stephan A. Missura. - [3]
A case study in w:Eiffel: [4]
psu: [5]
ieee: [6]
wikipedia: [7]
Uniform representation of basic algebraic structures in computer algebra [8], springer lecture notes in computer science, ISBN 978-3-540-61697-9
A Computer Algebra Solution to a Problem in Finite Groups [9]
python abstract algebra objects: [10]

infinity[edit]

Python has the finite set and list types already.
- Need: given a map between two sets, show that it is a bijection

What about infinite sets?
- Exercise: describe the category of finite-dimensional complex vector spaces in python

OR more precisely, we want to have a way to work with the category of groups (or a subcategory thereof) in the computer environment

What is a group? A group is a multiplet: (set, multiplication, inverse, identity).
how do we describe a group? simplest - by generators and relations; what else?
How are two groups related? Homomorphisms
- given a set map (which is a set of ordered pairs), we should construct a way to verify that it is a homomorphism - we can check (heuristically at least) that a map is a homomorphism by brute force computations on the generators
When do we know two groups are isomorphic?
- when we are given a bijective homomorphism
  - What if it is not given???

What are the operations on groups? direct product, quotient by a subgroup, semi-direct product, ... and a lot more

The theory of Lie groups, with its vast machinary of representation theory and with Dynkin diagrams as their DNAs, should be highly automatic!