Cova Algebra

A comprehensive Rust library for abstract algebra, providing rigorous implementations of algebraic structures from basic arithmetic to advanced category theory and tensor calculus.

Overview

Cova Algebra implements the fundamental structures of abstract algebra with a focus on mathematical correctness, type safety, and composability. The crate provides a hierarchical organization of algebraic concepts, from basic arithmetic operations to advanced constructions in algebra and category theory.

Architecture

The library is structured around core mathematical concepts, with each module building upon more fundamental structures:

Core Modules

arithmetic

Foundation layer providing basic arithmetic operations and modular arithmetic. Includes the modular! macro for creating custom modular number types and fundamental arithmetic traits that serve as building blocks for higher-level structures.

groups

Group theory implementations covering both commutative (Abelian) and non-commutative groups. Provides the fundamental structure for understanding symmetry and transformation in algebra, with proper distinctions between additive and multiplicative group operations.

rings

Ring theory abstractions including rings, fields, and semirings. Establishes the algebraic foundation for structures that support both addition and multiplication, with fields providing division operations for advanced algebraic computations.

modules

Module theory over rings, including vector spaces, semimodules, and specialized constructions like tropical modules. Provides the framework for linear algebra and generalizes vector spaces to work over arbitrary rings.

Advanced Modules

algebras

Higher-order algebraic structures that combine vector spaces with multiplication operations. Includes Boolean algebra for logical operations and Clifford algebras for geometric applications in physics and computer graphics.

tensors

Multi-dimensional tensor implementations with both compile-time fixed dimensions and runtime dynamic sizing. Supports tensor operations fundamental to linear algebra, differential geometry, and machine learning applications.

category

Category theory primitives providing abstract mathematical frameworks for composition and morphisms. Enables advanced mathematical constructions and provides a unifying language for describing mathematical structures and their relationships.

Design Principles

• Mathematical Rigor: All implementations follow strict mathematical definitions and maintain algebraic properties
• Type Safety: Leverages Rust's type system to encode mathematical constraints and prevent invalid operations
• Composability: Structures are designed to work together seamlessly, allowing complex mathematical constructions
• Performance: Balances mathematical correctness with computational efficiency through careful API design

Usage

Add to your Cargo.toml:

[dependencies]
cova-algebra = "*"

The crate provides a comprehensive prelude for convenient importing:

use cova_algebra::prelude::*;

Module Hierarchy

The algebraic structures follow a natural mathematical hierarchy:

Arithmetic Operations
    ├── Groups (symmetry and transformation)
    ├── Rings & Fields (number systems)
    └── Modules & Vector Spaces (linear structures)
        ├── Algebras (vector spaces with multiplication)
        ├── Tensors (multi-dimensional arrays)

Documentation

Complete API documentation is available on docs.rs.

Contributing

Contributions are welcome! Please ensure mathematical correctness and include appropriate documentation for any new algebraic structures.

License

This project is licensed under the AGPLv3 License - see the LICENSE file for details.