un_algebra
Simple Rust implementations of selected abstract algebraic structures.
Mathematical abstract algebra is built on a rich collection of algebraic structures. Learning about these structures can give non-mathematicians insights into the mathematical entities they need to work with--for example, real numbers, complex numbers, vectors, matrices, and permutations. By definition, these structures must comply with sets of axioms and properties, which are in turn a rich source of properties for generative testing.
un_algebra (_un_derstanding _algebra_) is a simple
implementation of selected algebraic structures in Rust. I hope it
is useful for developers learning abstract algebra concepts for the
first time. Currently this crate provides magma, semigroup,
quasigroup, monoid, group, ring and field implementations.
See https://docs.rs/un_algebra
Please refer to the [contributing] guide.
Add un_algebra to your Cargo.toml dependencies:
toml
[dependencies]
un_algebra = "0.2.1"
un_algebra is intended to support self-study of abstract algebraic
structures--it is not optimized for use in a production environment.
For production environments I recommend using a more sophisticated
library like [alga].
un_algebra uses 2018 edition features but unfortunately requires
Rust nightly as it uses the (experimental) external documentation
feature.
I'm not a mathematician so my implementation of the various
structures and their respective axioms in un_algebra may not be
strictly correct. Please let me know of any errors.
un_algebra implements the relevant structure traits for all the
Rust standard library integer and floating point types, for example,
an additive group for integer types i8, i16, i32, etc.
The Rust standard library has no support for complex numbers
(ℂ) or rational numbers (ℚ) so I've used the complex
and rational types from the [num] crate and implemented the
conforming traits in the [complex] and [rational] modules.
In addition, the crate examples directory contains abstract structure implementations of selected concepts, for example, finite fields.
Rust's planned i128 type forms several un_algebra algebraic
structures, starting with additive and multiplicative magmas (with
"wrapping" or modular arithmetic):
```rust use un_algebra::prelude::*;
impl AddMagma for i128 { fn add(&self, other: &Self) -> Self { self.wrapping_add(other) } }
impl MulMagma for i128 { fn mul(&self, other: &Self) -> Self { self.wrapping_mul(other) } } ````
i128 also forms additive and multiplicative semigroups:
```rust impl AddSemigroup for i128 {}
impl MulSemigroup for i128 {} ````
And additive and multiplicative monoids with one and zero as the monoid identities:
```rust impl AddMonoid for i128 { fn zero() -> Self { 0 } }
impl MulMonoid for i128 { fn one() -> Self { 1 } } ```
i128 also forms an additive group and additive commutative
group (with "wrapping" or modular negation), but not a
multiplicative group, as the integers have no closed division
operation:
```rust impl AddGroup for i128 { fn negate(&self) -> Self { self.wrapping_neg() } }
impl AddComGroup for i128 {} ```
And a ring and commutative ring:
```rust impl Ring for i128 {}
impl CommRing for i128 {} ```
Please refer to the [reading] document for more background on each structure and its associated axioms and properties.
Research other two-operation structures like semirings, division rings, and integral domains. Adding these might make the transition from groups up to fields more granular?
Research structures that combine two or more existing structures, for example modules or vector spaces.
The field traits probably need more testable derived properties.
Some structures have up to five trait variants: abstract, additive, multiplicative and their numeric counterparts. Research techniques for reducing the number of these traits with some kind of trait "context" that can be abstract, additive, or multiplicative.
Should the numeric equality traits be factored out to a separate
crate used by un_algebra?
Find a better method for embedding mathematical expressions in automatically generated Rust documentation.
Find an alternative approach for including external documentation so the crate can be built with a 2018 edition stable Cargo.
This project is licensed under the MIT license (see LICENSE.txt or https://opensource.org/licenses/MIT).