Simplify articles on groups and ring

[agryman]

The goal of the mathz project is to explore the usefulness of Z Notation for formalizing math. This project has been simmering for years and I'd like to make a decision about whether to persist with Z or to abandon it and use Lean 4. I do intend to use Lean 4 for proofs. I feel Z is much simpler and better integrated with LaTeX. However, formalizing even simple topics with Z can quickly get messy.

After starting a new article on rings and reviewing the current article on groups, I had several ideas for simplification of both. If these simplifications do not result in a more productive way to work then abandon Z for math and switch to Lean 4.

• The title of the groups should be changed to Algebraic Structures, or group-like algebraic structures
• Simplify the way I am using Z Notation by not relying so much on schemas - take advantage of tuples
• Eliminate the structure component from the schemas and simply form tuples as required
• Eliminate the generic prefix definitions. They don't improve readability much, aside from eliminating square brackets, but they add complexity since they required new LaTeX commands and Z definitions
• Try to use the globally defined mul operator, i.e (_ * _) within a schema by redefining it locally
• Use the technique used in the article on rings to eliminate multiple schemas that differ by renaming, i.e. avoid renaming by using decoration with primes
• Avoid awkward schema operators by mapping schemas to tuples that describe the structure but erase the component names.
• Schemas are good when we need to refer to variables
• Tuples are good for packaging components into structures.
• Abandon the practice of prefixing LaTeX command names.
  • This makes the names long and hard to type.
  • It is not a usual latex practice.
  • mathz is for my personal use.
  • It is just for taking notes to make sure I understand definitions.
  • No one else will use them.
  • The community will use Lean 4.

[agryman]

I dropped the mz prefix from the commands in the mathz-rings style.

I used the standard infix operators (+, -, *) as schema components.

I was unable to get fuzz to accept the unary minus as a schema component.

Next, simplify the groups article.