total.cppm File Reference

The Laws of Total Algebra (The Algebraic Hierarchy). More...

#include <algorithm>
#include <concepts>
#include <cstddef>
#include <functional>
#include <limits>
#include <ranges>
#include <type_traits>
#include <utility>
import dedekind.category:chain;

Include dependency graph for total.cppm:

Classes
struct	dedekind::category::is_periodic< T, Op >
struct	dedekind::category::is_saturating< T, Op >
struct	dedekind::category::is_exact_total< T, Op >
	Path D to totality: the operation is total because the carrier is exact — arithmetic with no rounding, Op(a,b) defined and lossless up to the carrier's out-of-memory tripwire. The archetypes are the exact number fields (Rational = ℚ, QuadraticReal = ℚ(√D)) under + and *. Distinct from the three machine-finite paths, which are total by staying in a bounded range (wrap / stabilise / saturate); an exact carrier is total because its arithmetic is closed and exact. More...
struct	dedekind::category::is_periodic< T, std::plus< T > >
struct	dedekind::category::is_periodic< T, std::minus< T > >
struct	dedekind::category::is_periodic< T, std::multiplies< T > >
struct	dedekind::category::is_periodic< T, std::bit_xor< T > >
struct	dedekind::category::construction_base< D >
	A carrier's construction: D is built from a base algebra and computes with the base's operations. ::type is the base; ::times_mixes_plus says whether D's × multiplies out (a quotient of a polynomial ring, the Cauchy product: the base's × and +) or stays componentwise (a product, a subalgebra: the base's × alone). Declared once per construction leg in dedekind.algebra (H / P / S / F), never per carrier; a carrier declaring two legs is ambiguous here, honestly. More...
struct	dedekind::category::is_total_by_construction< T, Op >
	Path E to totality: by construction. Defined below is_total, which it recurses into on the base. More...
struct	dedekind::category::is_total< T, Op >
	The certificate. Op is taken up to cv/ref (a const Inf keys the same registration as Inf). More...
struct	dedekind::category::is_total< T, detail_involution::Reversed< Op > >
	The opposite operation is total where the operation is, read off the value form, which sees registrations on the struct and on the value. More...
struct	dedekind::category::is_total< T, Op >
	A qualified operation is total where the operation is: the paths read the unqualified operation, and so does every specialisation. More...
struct	dedekind::category::is_total_by_construction< D, std::plus< D > >
struct	dedekind::category::is_total_by_construction< D, std::multiplies< D > >
struct	dedekind::category::unit_registry< ConstantMorphism< A, B >, Op >
	unit_registry specialisation for ConstantMorphism<A, B>. Registers the constant-at-identity-element-of-B as the identity element of ConstantMorphism<A, B> under operation Op. More...
struct	dedekind::category::is_cyclic_group< T, Op >
	Opt-in trait: is (T, Op) a \emph{cyclic} abelian group? More...
struct	dedekind::category::cyclic_order< T, Op >
	The order \(n\) of a cyclic group (its cardinality). More...
struct	dedekind::category::is_cyclic_group< T, std::plus< T > >
struct	dedekind::category::cyclic_order< T, std::plus< T > >

Namespaces
namespace	dedekind::category

Concepts
concept	dedekind::category::IsPeriodic
	Taxonomic Decorator: Certifies that an operation is circular (wraps). This is the "Safety Certificate" for machine totality.
concept	dedekind::category::IsSaturating
	Taxonomic Decorator: certifies that an operation reaches out-of-range values via a saturating escalation (e.g.\ \(\pm \aleph_0\)) rather than wrapping or being undefined.
concept	dedekind::category::IsClosedUnder
	(S, Op) is closed under a binary operation: Op{}(s_1, s_2) returns a value of type S for any S-typed operands. The textbook closure axiom expressed structurally — no opt-in trait needed because closure is deducible from the operator's signature.
concept	dedekind::category::IsClosedUnderUnary
	Unary closure variant: Op{}(s) returns S for any S-typed operand. Cardinality has no IsClosedUnderUnary for std::negate<> (negation of a positive natural would need to land outside ℕ); that's the structural witness that -Cardinality is closure-forcing rather than closed.
concept	dedekind::category::IsClosedUnderEither
	(T, Op) is closed under Op, where Op may be either binary or unary. Distinct from IsClosedUnder (binary) so call sites that need a specific arity stay typo-proof.
concept	dedekind::category::IsConstructed
	D declares a construction: a base and the shape of its ×.
concept	dedekind::category::CommutativeRingLaws
	The base's laws that a multiplied-out × rests on.
concept	dedekind::category::IsTotal
	The Master Safety Certificate for Level 0: is_total, as a concept.
concept	dedekind::category::IsTotalArrow
	A Morphism whose domain species admits no undefined (hazard) values.
concept	dedekind::category::IsPointed
	A type T has a chosen identity element under operation Op.
concept	dedekind::category::IsMagma
	T × T → T (The Base Total Species).
concept	dedekind::category::IsUnitalMagma
	A Magma with a global Identity (0 or 1).
concept	dedekind::category::IsInvertible
	A pointed (T, Op) where every element has an Op-inverse.
concept	dedekind::category::IsLoop
	A UnitalMagma with Inverses (-x).
concept	dedekind::category::IsSemigroup
	An Associative Magma.
concept	dedekind::category::IsCommutativeSemigroup
	An Associative Magma where the operation is also commutative.
concept	dedekind::category::IsMonoid
	A Semigroup with a two-sided Identity element.
concept	dedekind::category::IsMonoidHomomorphism
	A map between two monoids, (T, Op) and its image under Op2, that carries the operation, declared on every pair, and the unit to the unit, computed (:morphism's PreservesUnit).
concept	dedekind::category::IsCommutativeMonoid
	A Monoid where the operation is commutative.
concept	dedekind::category::IsGroup
	A Monoid with two-sided Inverses (The Perfect Symmetry).
concept	dedekind::category::IsAbelianGroup
	A Commutative Group.
concept	dedekind::category::IsCyclicGroup
	A cyclic abelian group: the honest concept name for \(\mathbb{Z}/n\mathbb{Z}\) carriers and their machine-backed cousins.
concept	dedekind::category::IsRig
	A Semiring without Negatives (Addition is a Monoid).
concept	dedekind::category::IsRng
	A Ring without Identity (Multiplication is a Semigroup).
concept	dedekind::category::IsSemiring
	Alias for IsRig: a general multi-operation species.
concept	dedekind::category::IsRing
	The Perfect Species (Rig ∩ Rng).
concept	dedekind::category::IsCommutativeRing
	A Ring whose multiplication is commutative.
concept	dedekind::category::IsField
	The axiomatic field witness — a commutative ring whose multiplicative structure is an abelian group (zero excluded, by convention on is_invertible_v).
concept	dedekind::category::IsJoinSemilattice
	A Semilattice used as the "join" (least upper bound / ∨) half of a Lattice. In bool this is logical_or; in int it is max.
concept	dedekind::category::IsMeetSemilattice
	A Semilattice used as the "meet" (greatest lower bound / ∧) half of a Lattice. In bool this is logical_and; in int it is min.
concept	dedekind::category::IsLattice
	compatibility alias for order-lattice refinement.
concept	dedekind::category::IsDistributiveLattice
	compatibility alias for distributive order lattices.
concept	dedekind::category::IsBoundedLattice
	A IsLattice whose join and meet each carry an identity: the bottom \(\bot\) (unit<T>(Join{})) and top \(\top\) (unit<T>(Meet{})).
concept	dedekind::category::IsBooleanAlgebra
	The top of the Boolean-algebra variety Alg(∨,∧,¬,⊥,⊤): a complemented distributive bounded lattice — the ∨/∧/¬ analogue of IsField topping the (signature-expanded) ring variety.
concept	dedekind::category::HasClosureForcingShape
	Structural shape: Op{} sends (A, A) → B (binary) or A → B (unary) with A ≠ B. Pure deducible clause of IsClosureForcing. The universal-property side — that B is the smallest carrier closing Op over A — is carried by the is_closure_forcing_v opt-in trait below.
concept	dedekind::category::IsClosureForcing
	Composite: structural shape + opt-in canonicality. The full witness that Op is the closure-forcing operator A → B (or A × A → B), with B the canonical recipient.
concept	dedekind::category::IsCopies
	A range of n copies of one value, Bird's replicate n x, whose fold under a monoid is the power \(x^n\); n and the value known without walking it: copies_count(r) and copies_value(r), which each model supplies, declared above so that ordinary lookup finds them for models outside this namespace, the standard's among them.

Typedefs
using	dedekind::category::XOR = std::bit_xor<unsigned int>
using	dedekind::category::AND = std::bit_and<unsigned int>

Functions
template<typename A, typename B, typename Op = std::plus<B>> requires IsPointed<B, Op>
constexpr auto	dedekind::category::zero ()
	zero<A, B, Op>(): the constant morphism A → B that maps everything to the Op-identity element of B. Defaults to std::plus (additive identity, i.e. zero); with std::multiplies it is the constant at one.
template<std::ranges::input_range R, typename T = std::ranges::range_value_t<R>, typename Op = std::plus<T>> requires IsMonoid<T, Op>
constexpr T	dedekind::category::concat (R r, Op op={})
	The free monoid's fold, concat, the canonical fold of a monoid: a range folded under the operation from its unit (the Jlt paper's Listings 2 to 4). Its specialisation, licensed by associativity, folds n copies of one value, any IsCopies range, by the binary method, one squaring per bit of n (Knuth, TAOCP 2 §4.6.3). Resolution picks it over the plain fold, its parameter being the more specific.
template<IsCountWord N, IsConstantArrow F>
constexpr N	dedekind::category::copies_count (const unfold_view< N, Mapped< N, StepCoalgebra< N, Prior< N > >, F > > &r)
	The counting chain of n under a constant arrow, n >> chain<N>() >> constant(x), is n copies of x, counted in the chain's own word.
template<IsCountWord N, IsConstantArrow F>
constexpr Cod< F >	dedekind::category::copies_value (const unfold_view< N, Mapped< N, StepCoalgebra< N, Prior< N > >, F > > &r)
template<std::copy_constructible T, std::integral B> requires (!std::same_as<B, bool>)
constexpr std::make_unsigned_t< B >	dedekind::category::copies_count (const std::ranges::repeat_view< T, B > &r)
	The standard's views::repeat(x, n) is n copies of x; its value is read only where n is not zero.
template<std::copy_constructible T, std::integral B> requires (!std::same_as<B, bool>)
constexpr T	dedekind::category::copies_value (const std::ranges::repeat_view< T, B > &r)
template<typename T, typename Op, IsCountWord N> requires IsMonoid<T, Op>
constexpr T	dedekind::category::detail_concat::power (const T &x, N n, Op op)
	\(x^n\) as the square of \(x^{\lfloor n/2 \rfloor}\), one x more where n is odd: the concat of the first half of the copies, used twice, since the halves are equal. A descent on n by halving to its base case, one call per bit of n, licensed by associativity.
template<std::ranges::input_range R, typename T = std::ranges::range_value_t<R>, typename Op = std::plus<T>> requires IsMonoid<T, Op> && IsCopies<R>
constexpr T	dedekind::category::concat (R r, Op op={})
	concat on n copies of one value: its n-th power, by squaring its first half, licensed by associativity. The same signature as the plain fold and a stronger constraint, so resolution prefers it wherever IsCopies holds. The value is read only where there is a copy, as a model promises it.

Variables
template<typename T, typename Op>
constexpr bool	dedekind::category::is_periodic_v = is_periodic<T, Op>::value
template<typename T, typename Op>
constexpr bool	dedekind::category::is_saturating_v = is_saturating<T, Op>::value
template<typename T, typename Op>
constexpr bool	dedekind::category::is_exact_total_v = is_exact_total<T, Op>::value
template<typename T, typename Op>
constexpr bool	dedekind::category::is_total_by_construction_v
	Value form of is_total_by_construction.
template<typename T, typename Op>
constexpr bool	dedekind::category::is_total_v = is_total<T, Op>::value
	Value form of is_total.
template<IsConstructed D>
constexpr bool	dedekind::category::is_associative_v< D, std::plus< D > >
template<IsConstructed D>
constexpr bool	dedekind::category::is_commutative_v< D, std::plus< D > >
template<IsConstructed D>
constexpr bool	dedekind::category::is_associative_v< D, std::multiplies< D > >
template<IsConstructed D>
constexpr bool	dedekind::category::is_commutative_v< D, std::multiplies< D > >
template<IsConstructed D>
constexpr bool	dedekind::category::is_distributive_v< D, std::multiplies< D >, std::plus< D > >
template<typename A, typename B, typename Op>
constexpr bool	dedekind::category::is_monotone_v< ConstantMorphism< A, B >, Op > = true
template<typename A, typename B, typename Op>
constexpr bool	dedekind::category::is_antimonotone_v< ConstantMorphism< A, B >, Op > = true
template<typename T, typename Op>
constexpr bool	dedekind::category::is_cyclic_group_v = is_cyclic_group<T, Op>::value
	Shorthand access to the is_cyclic_group trait.
template<typename T, typename Op>
constexpr std::size_t	dedekind::category::cyclic_order_v = cyclic_order<T, Op>::value
template<typename T, typename Join, typename Meet, typename Not>
constexpr bool	dedekind::category::is_complemented_v = false
	Opt-in complement law: T carries a complement Not w.r.t. the bounded lattice (Join, Meet), i.e. \(a \vee \neg a = \top\) and \(a \wedge \neg a = \bot\) for every a.
template<std::unsigned_integral T>
constexpr bool	dedekind::category::is_complemented_v< T, std::bit_or<>, std::bit_and<>, std::bit_not<> > = true
	The bit lattice of an unsigned word is complemented by ~: a | ~a is every bit, a & ~a none. Not bool, whose ~ promotes (~true is -2) and is no involution (std::logical_not is bool's complement).
template<>
constexpr bool	dedekind::category::is_complemented_v< bool, std::logical_or< bool >, std::logical_and< bool >, std::logical_not< bool > > = true
template<typename Op, typename A, typename B>
constexpr bool	dedekind::category::is_closure_forcing_v = false
	Opt-in trait carrying the universal-property clause of IsClosureForcing — the engineer's honesty obligation that B is the canonical (smallest) carrier closing Op over A inputs. C++ concepts cannot quantify universally, so this part of the witness is asserted, not deduced. Pattern mirrors is_associative_v / is_saturating.
constexpr std::size_t	dedekind::category::detail_concat_witness::ten = 10
constexpr std::size_t	dedekind::category::detail_concat_witness::seven = 7

Detailed Description

The Laws of Total Algebra (The Algebraic Hierarchy).

@partition :total

total__The_Categorical_Foundation

« Język jest aparatem wyznaczającym obraz świata. Struktura kategorii jest fundamentem wszelkiego poznania naukowego. » (Language is an apparatus that determines the image of the world. The structure of categories is the foundation of all scientific knowledge.) — Kazimierz Ajdukiewicz, 'Język i Poznanie' (Language and Cognition)

This partition defines the "Laws of Harmony" for a single species T under a binary operation Op. Following the Polish School of Logic, we treat these structures as species that "mature" as they gain axioms—moving from the primal Magma to the perfect symmetry of an Abelian Group.

total__Std_Namespace_Mappings

This partition asserts bidirectional mappings between algebraic hierarchy concepts and standard C++ types:

Std type	Highest algebraic structure
bool	Rig (OR/AND), Boolean Algebra (OR/AND/NOT)
unsigned int	Abelian Group (+), Ring (+,*), Ring (XOR,AND)
int	Distributive Lattice (max,min)
double	not a lattice: NaN breaks totality (see #934)

Copyright

2026 The Dedekind Authors

Note

"I obtained scientific guidance and stimulation mainly through personal mathematical contacts in Erlangen and in Gottingen. Above all I am indebted to E. Fischer, from whom I received the decisive impulse to study abstract algebra from an arithmetical viewpoint." – Emmy Noether, Habilitation curriculum vitae (1919)

See also

#275 nLab algebra/category hierarchy (work in progress): the quasigroup beside IsLoop.

Function Documentation

power()

template<typename T, typename Op, IsCountWord N>
requires IsMonoid<T, Op>

T dedekind::category::detail_concat::power ( const T & x, N n, Op op )

constexpr

\(x^n\) as the square of \(x^{\lfloor n/2 \rfloor}\), one x more where n is odd: the concat of the first half of the copies, used twice, since the halves are equal. A descent on n by halving to its base case, one call per bit of n, licensed by associativity.

Parameters

x	the value.
n	the copies.
op	the operation.

Returns

the concat of the n copies.