group.cppm File Reference

The Symmetry of Numbers (ℤ, ℚ*). More...

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

Include dependency graph for group.cppm:

Classes
struct	dedekind::algebra::Inverse< G, Op >
	The group inverse under Op, reified as an arrow (an endomorphism G → G). More...

Concepts
concept	dedekind::algebra::HasGroupOperatorsAdd
	Pure syntactic shape: T closes strictly under the additive-group operator surface +, binary -, unary -.
concept	dedekind::algebra::HasGroupOperatorsMul
	Pure syntactic shape: T closes strictly under the multiplicative-group operator surface *, /, and exposes the multiplicative unit T{1}.
concept	dedekind::algebra::HasSuccessorOperators
	Pure syntactic shape: T supports the successor / predecessor operator surface (pre- and post-increment / -decrement, all four).
concept	dedekind::algebra::HasCompoundGroupOperatorsAdd
	Pure syntactic shape: T supports the in-place additive-group operator surface +=, -=.
concept	dedekind::algebra::HasCompoundGroupOperatorsMul
	Pure syntactic shape: T supports the in-place multiplicative-group operator surface *=, /=.
concept	dedekind::algebra::IsGroup
	Proposition: a set object X whose carrier X::Domain forms a group — the set-indexed lift of category::IsGroup.
concept	dedekind::algebra::IsAdditiveGroup
	Proposition: a set object X whose carrier is a group under +.
concept	dedekind::algebra::IsMultiplicativeGroup
	Proposition: a set object X whose carrier is a group under *.
concept	dedekind::algebra::IsArithmeticAdditiveGroup
	The ℤ-deal: a carrier that is an additive group AND whose group structure agrees with the standard C++ +, -, unary - operators.

Detailed Description

The Symmetry of Numbers (ℤ, ℚ*).

@partition :group

Copyright

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

group__Taxonomy_of_Symmetry

This partition establishes the "Rules of Reflection" (Inverses). It promotes Monoids to Groups by verifying the existence of symmetric elements: the negative (-x) for addition and the reciprocal (1/x) for multiplication.

Note

"Det er utvilsomt at gruppeteorien er den mest fundamentale del av den moderne matematikken." ("It is beyond doubt that group theory is the most fundamental part of modern mathematics.") – Sophus Lie, Gesammelte Abhandlungen