cova-algebra 0.2.2

//! Group theory abstractions and implementations.
//!
//! This module provides traits and implementations for group theory concepts,
//! including both Abelian (commutative) and non-Abelian groups.

use super::*;

/// A trait representing a mathematical group.
///
/// A group is a set equipped with an operation that combines any two of its elements
/// to form a third element, satisfying four conditions called the group axioms:
/// closure, associativity, identity, and invertibility.
pub trait Group {
  /// Returns the identity element of the group.
  fn identity() -> Self;

  /// Returns the inverse of an element.
  fn inverse(&self) -> Self;
}

/// A trait representing an Abelian (commutative) group.
///
/// An Abelian group is a group where the group operation is commutative.
/// This trait combines the requirements for a group with additional operations
/// that are natural for commutative groups. We mark this as an [`Additive`] structure since this is
/// typical notation for Abelian groups.
pub trait AbelianGroup:
  Group + Zero + Additive + Neg<Output = Self> + Sub<Output = Self> + SubAssign {
}

/// A trait representing a non-Abelian group.
///
/// A non-Abelian group is a group where the group operation is not necessarily commutative.
/// This trait combines the requirements for a group with additional operations
/// that are natural for non-commutative groups. We mark this as a [`Multiplicative`] structure
/// since this is typical notation for non-Abelian groups. However, it should be noted that a
/// [`NonAbelianGroup`] group cannot be an [`AbelianGroup`]
pub trait NonAbelianGroup: Group + One + Multiplicative + Div<Output = Self> + DivAssign {}