alga2

A modern abstract-algebra hierarchy for Rust — the successor to alga (unmaintained since 2020), powered by batch-impl.

~900 impls generated from ~80 batch-impl DSL blocks — each method body written once, the matrix supplying the quantity.

Quick start

[dependencies]
alga2 = "0.1"

use alga2::op::{Additive, Multiplicative};
use alga2::tower::{DivisionRing, Group, Magma, Monoid};

// One type, two operators: `u8` is a monoid under both `+` and `*`.
assert_eq!(<u8 as Magma<Additive>>::combine(&3, &4), 7);
assert_eq!(<u8 as Magma<Multiplicative>>::combine(&3, &4), 12);
assert_eq!(<u8 as Monoid<Additive>>::identity(), 0);
assert_eq!(<u8 as Monoid<Multiplicative>>::identity(), 1);

// The additive ladder reaches groups: inverse and field inverse.
assert_eq!(<i32 as Group<Additive>>::inverse(&5), -5);
assert_eq!(<f64 as DivisionRing<Additive, Multiplicative>>::inv(&2.0), 0.5);

The hierarchy

The tower traits are parameterized over operator markers (Additive / Multiplicative), so one type can inhabit each level twice without colliding:

• additive ladder: Magma → Semigroup → Monoid → Group → AbelianGroup (with the parallel Quasigroup → Loop leg — a group is an associative loop; (R, ·) is not a quasigroup, so the multiplicative side stops at Monoid)
• multiplicative ladder: Magma → Semigroup → Monoid
• semiring ladder: Semiring → Ring → CommutativeRing → Field (with the non-commutative DivisionRing — the quaternions are its classic inhabitant, provided in-crate)
• module level: Module (a ring acting on an abelian group) and VectorSpace (scalars form a field) — every numeric is a module over itself; tuples of same-scalar modules are modules, componentwise
• extended structures: StarSemiring (Kleene star), Band (idempotent semigroups), EuclideanDomain (euclidean division + gcd), LieAlgebra (bracket), Power (square-and-multiply)
• field refinements: OrderedField, FiniteField (with bool = F₂ as inhabitant), ComplexField, FieldExtension/FieldExtensionTower (C is a degree-2 extension of R, in-crate), FreeModule (the f64 tuples are Rⁿ)
• analytic level: Lattice (meet/join), SubsetOf/SupersetOf (numeric embeddings), ClosedAdd-family markers, Real, BilinearForm/ SymmetricBilinearForm/PositiveDefinite (the multiplication form on the reals is the canonical example), TensorProduct (bilinearity laws), and the normed layer — NormedSpace/InnerSpace/FiniteDimVectorSpace/ FiniteDimInnerSpace (Gram–Schmidt included), plus EuclideanSpace/ AffineSpace and the Matrix/Transformation interfaces (Isometry/DirectIsometry/OrthogonalTransformation, InversibleSquareMatrix; alga-aligned, for downstream geometry types)

Integer operations are wrapping (mod 2^N): under Additive, u8 is the group Z/256Z, which plain + would break with an overflow panic in debug builds.

Law testing (laws)

Every level ships proptest properties and law bundles, so downstream users check a custom type against the laws of the level it claims to implement (alga2::laws, feature proptest):

// Requires the `proptest` feature; run inside your own test module.
use alga2::laws::ring_laws;
use proptest::prelude::*;

proptest! {
    fn my_type_obeys(a: MyType, b: MyType, c: MyType) {
        ring_laws(a, b, c)?;
    }
}

Built with batch-impl

Every impl in this crate is generated by batch-impl — a proc-macro DSL for generating trait impls in bulk. The whole ~900-impl matrix is ~80 DSL blocks; each method body is written exactly once, and the matrix supplies the quantity (types × operators × arities). The few genuinely bulky algorithms (the Hamiltonian product, extended euclid) are hand-written and clearly marked.

alga2 is batch-impl's flagship showcase: the DSL idioms this crate relies on (trait-name inheritance, in-constraint @trait references, shape templates, X<> operator sync, variadic tuple ranges) were driven by alga2's needs and landed upstream. The patterns are documented in docs/DESIGN.md; if batch-impl can express a matrix that interests you, this crate is the working proof.

Layout

• tower — the trait hierarchy (single source of truth)
• op — the operator markers
• laws — proptest law testing (feature proptest)
• complex — in-crate Complex<T>, Quaternion<T> (zero-dependency)
• impls (private) — the batch-impl matrices

The core is no_std; container impls layer on via alloc and std (default) features. Design notes: docs/ROADMAP.md.