dedekind::morphologies::IsCyclic Concept Reference

\emph{Operational} duck-typed check that T carries the member API of a cyclic structure: T::Domain alias, T::generator() returning a Domain element, and T::successor(a) walking the chain. More...

Concept definition

template<typename T>
concept dedekind::morphologies::IsCyclic =  requires {
  typename T::Domain;
  { T::generator() } -> std::same_as<typename T::Domain>;
} && requires(typename T::Domain a) {
  { T::successor(a) } -> std::same_as<typename T::Domain>;
}

Detailed Description

\emph{Operational} duck-typed check that T carries the member API of a cyclic structure: T::Domain alias, T::generator() returning a Domain element, and T::successor(a) walking the chain.

This is the syntactic counterpart of the categorical dedekind::category::IsCyclicGroup<T, Op> in the strict / operational pattern the project uses elsewhere (cf. HasRingOperators vs IsRing). The two concepts are complementary:

• IsCyclic<T> — "this carrier exposes the shape" (member API for generator + successor). Cheap to check; useful for duck-typed templated algorithms that walk a cyclic chain.
• category::IsCyclicGroup<T, Op> — "this carrier under this operation forms a cyclic abelian group" (axiomatic, via IsAbelianGroup<T, Op> + the is_cyclic_group_v<T, Op> opt-in trait). More demanding; certifies the laws.

Carriers that satisfy both — e.g. morphologies::Modular<N> — are positively certified at both layers. Other carriers, such as morphologies::CyclicRing<T, N>, may satisfy only the operational layer unless separately registered with the categorical trait machinery.

archimedean__Vocabulary_Notes_388

The T::successor(x) member API exposed by IsCyclic carriers coincides numerically with the Peano successor \(S(x) = x + 1\) but is a different beast in the type system from the axiomatic dedekind::order::IsSuccessor<T> concept (in dedekind.order:completeness). The pairing is:

• Carrier-level: a static member T::successor(x) returning a Domain element. Operational; what the IsCyclic shape concept reaches for when walking a cyclic chain.
• Concept-level: order::IsSuccessor<T>, asserting x + unit<T>(std::multiplies<T>{}) is well-typed and that T is a partial magma + multiplicatively pointed. Axiomatic; the trait the order::IsArchimedean composition demands.

The two coincide on carriers like Modular<N> that satisfy both surfaces. The coherence witness Modular<N>::successor(x) == x + Modular<N>{1} is pinned at the type level in src/main/modules/dedekind/morphologies/cyclic.cppm (alongside the IsCyclic / IsCyclicGroup chains for the same carrier). Re-expressing order::IsArchimedean purely in terms of :order concepts is deliberately not done here: the additive partial-magma + pointed multiplicative-identity composition is the algebraic content the Archimedean axiom genuinely needs, and the order layer is the right home for it.

archimedean__Modern_Bridge_Archimedean_Generators

The Peano successor and the Archimedean property are unified in modern probability through the Archimedean generator of copula theory: a strictly decreasing convex function \(\varphi : [0, 1] \to [0, \infty]\) with the additive law

\[ \varphi(C(u, v)) \;=\; \varphi(u) \,+\, \varphi(v), \]

where C is an Archimedean copula. Two threads of the library meet in this framing:

• Peano: the successor recursion \(S(n) = n + 1\) is the finite analogue of \(\varphi\)'s additive layer-stacking under C. Adding 1 once corresponds to composing one more dependency layer via \(\varphi^{-1}\).
• Archimedean: the order-theoretic scale axiom (every element reachable from a base point by finite iteration of S) is the discrete analogue of the copula-theoretic d-monotonicity condition — \(\varphi\)'s pseudo-inverse must be "sufficiently large" for the dependency to scale across any number of variables.

This is the deeper motivation behind the project's twin successor / generator vocabulary: the IsCyclic carrier supplies a discrete generator g and a successor walk g, S(g), S(S(g)), \ldots; an order::IsArchimedean carrier supplies the same shape at the axiom level; an Archimedean copula lifts the same shape into the continuous unit interval. A dedicated :copula partition is out of scope here, but the framing is recorded so future work can connect the discrete and continuous Archimedean stories without re-deriving the bridge.