An arrow \(f : A \to B\) between two algebras that carries each operation of the signature S on \(A\) to the one in the same place of T on \(B\): \(f(x \odot y) = f(x) \circledast f(y)\) and \(f(e) = e'\) for each (Burris and Sankappanavar, A Course in Universal Algebra, §II.6). More...

Concept definition

template<typename Arrow, typename S typename detail_universal::default_signatures<Arrow>::source, typename T typename detail_universal::default_signatures<Arrow>::target>
concept dedekind::algebra::IsHomomorphism = 
    dedekind::category::IsArrow<Arrow> &&
    detail_universal::homomorphism_of<std::remove_cvref_t<Arrow>, S, T>::value

Detailed Description

An arrow \(f : A \to B\) between two algebras that carries each operation of the signature S on \(A\) to the one in the same place of T on \(B\): \(f(x \odot y) = f(x) \circledast f(y)\) and \(f(e) = e'\) for each (Burris and Sankappanavar, A Course in Universal Algebra, §II.6).

Per operation the law is the declared certificate category::is_homomorphism, keyed by the arrow, its domain and the pair of operations, of any arity, and where the operation has a unit, the law at the unit is computed, category::PreservesUnit. The signatures default to the semiring operations each carrier registers, semiring_ops, so a ring embedding is named by its arrow alone; an empty signature asserts nothing and is refused, and so is a type that is no arrow. The honest discipline is to pin each declaration with computed witnesses at its site.

Template Parameters

Arrow	the arrow.
S
T	the signatures of its domain and codomain.

See also: Lang, Algebra (3rd ed.), §III.1 (homomorphisms of rings); Mac Lane, Categories for the Working Mathematician §III.1 (homomorphisms as morphisms in the category of algebras).