dedekind::algebra::IsOrderedCommutativeRing Concept Reference

A commutative ring whose carrier is ordered under the additive translation marker. More...

Concept definition

template<typename T>
concept dedekind::algebra::IsOrderedCommutativeRing = 
    dedekind::category::IsCommutativeRing<T, std::plus<T>,
                                          std::multiplies<T>> &&
    is_translation_invariant_ordered_v<T>

Detailed Description

A commutative ring whose carrier is ordered under the additive translation marker.

Composes:

• category::IsCommutativeRing<T, +, *> — T is an axiomatic commutative ring.
• is_translation_invariant_ordered_v<T> — T's order is compatible with the additive group (re-use of the additive marker; in a commutative ring with ordered addition, positive-scalar multiplication automatically preserves order and negative-scalar multiplication automatically reverses).

ℤ (SignedCardinality) satisfies this; ℚ also does (a field is a fortiori a commutative ring), and the two are distinguished by the stronger IsOrderedMultiplicativeGroup gate where only the ring structure is available.