Finite Algebra Primitives

The integration surface where the finite capability meets the algebra classifications: a cyclic-group (ℤ/Nℤ) witness for every Finite.Enumerable, plus Finite.Enumerable conformances for the Parity and Polarity classifications.

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Quick Start

Any N-element Finite.Enumerable is isomorphic to the cyclic group ℤ/Nℤ: each value maps to an integer in 0..<N via its ordinal, and the group operation is addition modulo N. This package confers that structure on every conformer through a single static accessor — Algebra.Group<Element>.cyclic — without reimplementing the group per type.

import Finite_Algebra_Primitives

// Parity is a 2-element Finite.Enumerable, so its cyclic group is ℤ/2ℤ.
let z2 = Algebra.Group<Parity>.cyclic

z2.identity            // .even   — the value at ordinal 0
z2(.even, .odd)        // .odd    — 0 + 1 mod 2
z2(.odd, .odd)         // .even   — 1 + 1 mod 2
z2.inverting(.odd)     // .odd    — every element is self-inverse in ℤ/2ℤ

The same accessor scales to any cardinality. Polarity has three values (positive, negative, neutral), so it yields ℤ/3ℤ:

import Finite_Algebra_Primitives

Polarity.count                  // 3
Polarity.negative.ordinal       // 1

let z3 = Algebra.Group<Polarity>.cyclic

z3(.negative, .negative)        // .neutral    — 1 + 1 mod 3 = 2
z3(.negative, .neutral)         // .positive   — 1 + 2 mod 3 = 0
z3.inverting(.negative)         // .neutral    — 3 − 1 mod 3 = 2

Importing Finite_Algebra_Primitives re-exports all four bridged primitives — the Finite capability, the Algebra and Algebra.Group machinery, and the Parity / Polarity classifications — so the conformances and the group are visible from one import.

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Installation

dependencies: [
    .package(url: "https://github.com/swift-primitives/swift-finite-algebra-primitives.git", branch: "main")
]

.target(
    name: "App",
    dependencies: [
        .product(name: "Finite Algebra Primitives", package: "swift-finite-algebra-primitives"),
    ]
)

Requires Swift 6.3.1 and macOS 26 / iOS 26 / tvOS 26 / watchOS 26 / visionOS 26 (or the matching Linux / Windows toolchain).

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Architecture

One library product plus its test-support spine. This package adds no new types: it bridges four existing primitives — swift-finite-primitives, swift-algebra-primitives, swift-parity-primitives, and swift-polarity-primitives — and re-exports them.

Product	Target	Purpose
Finite Algebra Primitives	Sources/Finite Algebra Primitives/	The integration surface: Algebra.Group<Element>.cyclic (the ℤ/Nℤ group over any Finite.Enumerable), Finite.Enumerable conformances for Parity and Polarity, and a re-export of the four bridged primitives.
Finite Algebra Primitives Test Support	Tests/Support/	Re-exports the main target and the upstream algebra test support for test consumers.

Foundation-free.

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Platform Support

Platform	Status
macOS 26	Full support
Linux	Full support
Windows	Full support
iOS / tvOS / watchOS / visionOS	Supported

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Community

License

Apache 2.0. See LICENSE.md.