lattice.cppm File Reference

Geometric lattices (ℤ², ℕ², Gaussian integers in ℂ). More...

#include <concepts>
#include <utility>
import dedekind.sets;
import dedekind.category;
import dedekind.geometry:affine;

Include dependency graph for lattice.cppm:

Classes
struct	dedekind::geometry::LatticeFactory< NaturalLatticeSet >
struct	dedekind::geometry::LatticeFactory< 𝔸< I > >

Namespaces
namespace	dedekind::category

Concepts
concept	dedekind::geometry::IsGeometricLattice
	A Set S whose elements are integer-coordinate points, embedded into a continuous vector space V via a declared-monic arrow E.

Typedefs
using	dedekind::geometry::NaturalLatticeSet = 𝔸<Cardinality>
using	dedekind::geometry::IntegerLatticeSet = 𝔸<int>
using	dedekind::geometry::IntegerLatticeScalar = typename IntegerLatticeSet::Domain
using	dedekind::geometry::IntegerLatticePoint2D
using	dedekind::geometry::NaturalLatticeScalar = Cardinality
using	dedekind::geometry::NaturalLatticePoint2D

Functions
constexpr auto	dedekind::geometry::natural_lattice_2d ()
	The unbounded natural lattice in 2D: Λ_nat = { (x, y) ∈ ℕ² }.
constexpr auto	dedekind::geometry::integer_lattice_2d ()
	The unbounded integer lattice in 2D: Λ_int = { (x, y) ∈ ℤ² }.
constexpr auto	dedekind::geometry::square_natural_grid (unsigned int n)
	The canonical N×N square natural grid: G_N^nat = { (x, y) ∈ ℕ² | 0 ≤ x < n, 0 ≤ y < n }.
constexpr auto	dedekind::geometry::square_integer_grid (IntegerLatticeScalar lower, IntegerLatticeScalar upper)
	Bounded square integer grid: G_Z = { (x, y) ∈ ℤ² | lower ≤ x < upper, lower ≤ y < upper }.

Variables
template<typename S>
constexpr LatticeFactory< S >	dedekind::geometry::lattice {}
constexpr auto	dedekind::geometry::embed_z2_r2
	The canonical embedding ℤ² ↪ ℝ²: (x, y) ↦ Vector{x, y}.
template<>
constexpr bool	dedekind::category::is_monic_arrow_v< std::decay_t< decltype(dedekind::geometry::embed_z2_r2)> >

Detailed Description

Geometric lattices (ℤ², ℕ², Gaussian integers in ℂ).

@partition :lattice

Copyright

2026 The Dedekind Authors Licensed under the Apache License, Version 2.0.

Note

"解伏題之法は、消去の術に在り" — Seki Kōwa (関 孝和, 1642–1708), paraphrase of the central thesis of Kaifukudai no Hō (解伏題之法, 1683), the treatise that introduced determinant theory roughly a decade before Leibniz. [Trans: "The method of solving buried problems lies in the technique of elimination."]

Seki computed on a sangi (算木) wooden board — a literal integer lattice — making his school's algebraic technique a natural ancestor of the IntegerLatticePoint2D / integer_lattice_2d() carriers below.

Typedef Documentation

IntegerLatticePoint2D

using dedekind::geometry::IntegerLatticePoint2D

export

Initial value:

 
    std::pair<IntegerLatticeScalar, IntegerLatticeScalar>

NaturalLatticePoint2D

using dedekind::geometry::NaturalLatticePoint2D

export

Initial value:

 
    std::pair<NaturalLatticeScalar, NaturalLatticeScalar>

Function Documentation

square_integer_grid()

auto dedekind::geometry::square_integer_grid ( IntegerLatticeScalar lower, IntegerLatticeScalar upper )

constexprexport

Bounded square integer grid: G_Z = { (x, y) ∈ ℤ² | lower ≤ x < upper, lower ≤ y < upper }.

Uses the ambient integer lattice species and explicit bounds, so negative coordinates are admitted when lower < 0.

square_natural_grid()

auto dedekind::geometry::square_natural_grid ( unsigned int n )

constexprexport

The canonical N×N square natural grid: G_N^nat = { (x, y) ∈ ℕ² | 0 ≤ x < n, 0 ≤ y < n }.

Built as the cartesian product of two finite bounded naturals using the set-builder DSL. This is the lattice domain before any continuous embedding. Post-#401 the grid coordinates are NaturalLatticePoint2D = pair<unsigned, unsigned> to match the ℕ carrier reading.

Parameters

n	Side length of the grid (number of points per axis).

Returns

A Set<NaturalLatticePoint2D, Kleene, ...>.

Variable Documentation

embed_z2_r2

auto dedekind::geometry::embed_z2_r2

inlineconstexprexport

Initial value:

=
    arrow<IntegerLatticePoint2D, Vector<double, 2>>(
        [](const IntegerLatticePoint2D& p) noexcept -> Vector<double, 2> {
          return {static_cast<double>(p.first), static_cast<double>(p.second)};
        })

The canonical embedding ℤ² ↪ ℝ²: (x, y) ↦ Vector{x, y}.

Maps each integer pair to the corresponding point in ℝ². This is a group homomorphism (ℤ², +) → (ℝ², +). Declared monic below: distinct integer pairs map to distinct vectors since component equality implies coordinate equality.