The MULTIPLICATIVE order-compatibility axiom marker (O2): the positive cone is closed under *, i.e. 0≤a ∧ 0≤b ⟹ 0≤a·b — equivalently, scaling by a positive preserves the order. The multiplicative sibling of is_translation_invariant_ordered (O1, scaling ↔ translation).
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Detailed Description
struct dedekind::algebra::is_scaling_invariant_ordered< T >
The MULTIPLICATIVE order-compatibility axiom marker (O2): the positive cone is closed under *, i.e. 0≤a ∧ 0≤b ⟹ 0≤a·b — equivalently, scaling by a positive preserves the order. The multiplicative sibling of is_translation_invariant_ordered (O1, scaling ↔ translation).
Defaults false_type; a carrier opts in only when its order is genuinely compatible with * (an ordered field). This cannot be a concept check: O2 is a \(\forall\)-law over VALUES, and a concept inspects only types / operation signatures — the same wall that makes is_invertible_v / is_associative_v opt-in traits, not concept checks.
- Warning
- An earlier design leaned on
order::IsTotallyOrdered<T>to stand in for this axiom ("a separate marker would only re-state what the upstream concepts prove"). That is FALSE:std::totally_orderedis mere syntactic comparability and does NOT prove O2. \(\mathbb{F}_5\) orders its representatives0<1<2<3<4(so it isstd::totally_ordered) yet1·3=3> 2·3=1 flips the order and−1=2²is a square — it is a field that is NOT formally real, andComplex<𝔽₅> splits into zero divisors. The structural<=>check belongs at this opt-in site (per carrier), not in the concept — exactly as for the additive marker.
The documentation for this struct was generated from the following file:
- src/main/modules/dedekind/algebra/ordered_algebra.cppm
