## nightly free-algebra

Types for constructing free algebras over sets

### 1 unstable release

 0.1.0 Nov 9, 2019

#1561 in Math

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Types for constructing free algebras over sets in the Rust programming language.

## What even is a "Free Algebra"?

In the context of this crate, the term "Algebra" refers to a range of mathematical constructions involving arithmetic operations, and the term "Free" refers to the nature of those operations. In particular, a "free" operation is one that is made with as little restriction as possible with respect to the desired set of rules.

So, in general, the procedure for such a "free" construction is to start with some type `T` and some set of algebraic rules, and then operated on the elements of `T` as if they were a variable or symbol, applying the rules as necessary.

As abstract as that sounds, there is actually a prime example of this already in the standard library, the `Vec<T>`! If we start with some type `T`, assert that multiplication be associative, and start multiplying elements like variables, the result is exactly the same as if we took `Vec<T>` and implemented multiplication as concatenation. In fact, this is precisely what the `FreeMonoid<T>` type in this crate is.

``````use maths_traits::algebra::One;
use free_algebra::FreeMonoid;

let x: FreeMonoid<char> = FreeMonoid::one();
let y = FreeMonoid::one() * 'a' * 'b';
let z = FreeMonoid::one() * 'c' * 'd';

assert_eq!(x, vec![]);
assert_eq!(y, vec!['a', 'b']);
assert_eq!(z, vec!['c', 'd']);
assert_eq!(&y * &z, vec!['a', 'b', 'c', 'd']);
assert_eq!(&z * &y, vec!['c', 'd', 'a', 'b']);

``````

In addition to this, moreover, a number of other constructions can be achieved by changing which types are used, which operations are considered, and what rules are followed. Examples include:

• `FreeModule<R,T>`: Results from freely adding elements of `T` in an associative and commutative manner and allowing distributive multiplication by elements from `R`
• `FreeAlgebra<R,T>`: The same as with `FreeModule`, except that we allow for free multiplication of elements distributively (like with `FreeMonoid`)
• Polynomials: A `FreeAlgebra`, but where multiplication between `T`'s is commutative and associative
• Clifford algebra: A `FreeAlgebra`, but where multiplication is associative and an element times itself results in a scalars
• Complex numbers: Results from when `T` is either `1` and `i` and multiplies accordingly
• Quaternions: Same as for Complex numbers, but with more imaginary units

## Use cases

The primary purposes for this crate fall into two general categories:

• Use as an abstract foundation to create more specific systems like polynomials or Clifford algebras.
• Utilization as a tool for lazily storing costly arithmetic operations for future evaluation.

## Crate structure

This crate consists of the following:

• Two main structures for doing the free-arithmetic over some type
• Traits for specifying the rules for arithmetic
• Type aliases for particular combinations of construction and rules

Specifically:

• `MonoidalString` constructs free-multiplying structures over a type `T` using an order-dependent internal representation with a `Vec<T>` that determines its multiplication rule using an implementor of the trait `MonoidRule`. Aliases of this struct include `FreeMonoid` and `FreeGroup`.
• `ModuleString` constructs types consisting of terms of type `T` with scalars from some additive type `R` stored with an order independent `HashMap`. This grants all `ModuleString`'s an addition operation by adding the coefficients of like terms, and a free-multiplication can be included using an optional `AlgebraRule` parameter. Aliases of this struct include `FreeModule` and `FreeAlgebra`.