dedekind::algebra::IsField Concept Reference

The set-indexed lift of the axiomatic category::IsField: a set object X whose carrier X::Domain is a field. More...

Concept definition

template<typename X, typename Add std::plus<typename X::Domain>, typename Mult std::multiplies<typename X::Domain>>
concept dedekind::algebra::IsField =  IsAlgebraOnSet<X, Add, Mult> &&
                  dedekind::category::IsField<typename X::Domain, Add, Mult>

Detailed Description

The set-indexed lift of the axiomatic category::IsField: a set object X whose carrier X::Domain is a field.

Template Parameters

X	The set object (category::IsSet).
Add	Additive operation witness (defaults to std::plus).
Mult	Multiplicative operation witness (defaults to std::multiplies).

The faithful lift, matching every other rung: IsAlgebraOnSet supplies the carrier discipline (IsLSet roots std::semiregular on X::Domain, IsClosedAlgebra pins strict closure) and category::IsField<X::Domain, Add, Mult> supplies the axioms — a commutative ring whose non-zero elements are multiplicatively invertible (via is_invertible_v / IsAbelianGroup<X::Domain, Mult>).

Note

The division operator surface (operator/, .inverse(), std::divides) is deliberately not required here. It is a syntactic concern carried by the separate concepts HasFieldOperators (shape only) and IsDivisionRing (ring + surface). Keeping it out makes IsField a pure axiomatic lift, so a genuine field with no division operator — \(\mathbb{F}_2 = \mathbb{A}\langle\texttt{bool}\rangle\) under (bit_xor, bit_and) — qualifies. A callsite that needs to write a/b on field elements constrains additionally on HasFieldOperators.