The Rules of Harmony (The Semiring Synthesis). More...

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

Include dependency graph for ring.cppm:

Concepts
concept	dedekind::algebra::HasRingOperators
	Pure syntactic shape: T closes strictly under `+`, `-`, `*` (binary and unary `-`). Textbook closure — results are exactly `T`, not just convertible to `T`.
concept	dedekind::algebra::IsSemiring
	The set-indexed lift of `category::IsSemiring` (an exact alias of `category::IsRig`): the "rig" rung of the tower.
concept	dedekind::algebra::IsRig
concept	dedekind::algebra::IsRng
concept	dedekind::algebra::IsRing
	The set-indexed lift: a set object `X` whose carrier `X::Domain` is both a Semiring AND an Abelian Group under addition.
concept	dedekind::algebra::IsArithmeticRing
	The seal: a carrier that is a ring AND whose ring structure agrees with the standard C++ arithmetic operators.
concept	dedekind::algebra::IsCommutativeRing
	A Ring where multiplication is also commutative.
concept	dedekind::algebra::HasSemiringOperators
	Pure syntactic shape: T closes strictly under `+` and `*`, with structural identities `T{}` and `T{1}`.
concept	dedekind::algebra::IsRingHomomorphism
	A callable φ from `Source` to `Target` that carries the ring structure: \(\varphi(a + b) = \varphi(a) + \varphi(b)\), \(\varphi(a b) = \varphi(a) \varphi(b)\), \(\varphi(0) = 0\) and \(\varphi(1) = 1\) (Lang, Algebra, §II.1).

Detailed Description

The Rules of Harmony (The Semiring Synthesis).

@partition :ring

Note: „Was beweisbar ist, soll in der Wissenschaft nicht ohne Beweis geglaubt werden.“ (What is provable should not be believed without proof.) — Richard Dedekind

Concepts