Semirings as the formal algebraic basis for scalar number systems. More...

#include <concepts>
#include <functional>
import dedekind.sets;
import dedekind.category;
import dedekind.algebra;

Include dependency graph for scalars.cppm:

Namespaces
namespace	dedekind::numbers

Concepts
concept	dedekind::numbers::IsIntegralScalar
	Numbers-layer alias for the algebra scalar bridge concept.
concept	dedekind::numbers::IsUnsignedIntegralScalar
	Numbers-layer alias for the algebra scalar bridge concept.
concept	dedekind::numbers::IsFloatingScalar
	Numbers-layer alias for the algebra scalar bridge concept.
concept	dedekind::numbers::IsScalar
concept	dedekind::numbers::IsNumbers
	The Root Category for all Numerical Structures.

Detailed Description

Semirings as the formal algebraic basis for scalar number systems.

@partition :scalars @dependency :algebra, :topology, :cardinalities

scalars__Numbers

This partition defines the Scalar—the fundamental unit of the ontology. It maps concrete C++ types to their formal algebraic and topological identities.

A Scalar is the "Lowest Common Denominator" of our number systems. By grounding the definition in a Semiring (or "Rig"), we create a unified interface that encompasses:

Booleans : The Binary Semiring ({0, 1}, ∨, ∧).
Naturals : The Discrete Monoid (ℕ, +, ×).
Reals : The Continuous Field (ℝ, +, ×).

This structure guarantees an additive identity (Zero) and a multiplicative identity (One), providing the necessary anchors for all coordinate-based calculations without requiring additive inverses (negatives).

@anchors C++ Fundamental Types: bool, char, int, long, float, double.

Wikipedia: Scalar (mathematics), Semiring, Monoid, Number system

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

Note: "Das Wesen der Mathematik liegt gerade in ihrer Freiheit." ("The essence of mathematics lies precisely in its freedom.") – Georg Cantor

Namespaces

Concepts

Detailed Description

scalars__Numbers