A callable φ from Source to Target that carries the ring structure: \(\varphi(a + b) = \varphi(a) + \varphi(b)\), \(\varphi(a b) = \varphi(a) \varphi(b)\), \(\varphi(0) = 0\) and \(\varphi(1) = 1\) (Lang, Algebra, §II.1). More...

Concept definition

template<typename Phi, typename Source, typename Target>
concept dedekind::algebra::IsRingHomomorphism = 
    dedekind::category::IsRing<Source, typename semiring_ops<Source>::add,
                               typename semiring_ops<Source>::mult> &&
    dedekind::category::IsRing<Target, typename semiring_ops<Target>::add,
                               typename semiring_ops<Target>::mult> &&
    dedekind::category::IsArrow<Phi> &&
    std::same_as<dedekind::category::Dom<Phi>, Source> &&
    std::same_as<dedekind::category::Cod<Phi>, Target> &&
    IsHomomorphism<Phi, semiring_signature<Source>, semiring_signature<Target>>

Detailed Description

A callable φ from Source to Target that carries the ring structure: \(\varphi(a + b) = \varphi(a) + \varphi(b)\), \(\varphi(a b) = \varphi(a) \varphi(b)\), \(\varphi(0) = 0\) and \(\varphi(1) = 1\) (Lang, Algebra, §II.1).

A homomorphism of the semiring signatures, IsHomomorphism of the (⊕, ⊗) each ring registers on semiring_ops: two monoid homomorphisms, the laws on every pair declared on category::is_homomorphism and the laws at the units computed. The negation comes along, since a monoid homomorphism between additive groups is a group homomorphism. An arrow between two rings, each certified category::IsRing under the operations it registers.

Template Parameters

Phi	the map.
Source
Target	the rings.