Subobject S of A is a subalgebra under operation Op when it is closed under Op. More...

Concept definition

template<typename S, typename A, typename Op>
concept dedekind::category::IsSubalgebra =  IsSubobject<S, A> && is_closed_under_v<S, A, Op> &&
                       requires(const A& a, const Op& op) {
                         { op(a, a) } -> std::convertible_to<A>;
                       }

Detailed Description

Subobject S of A is a subalgebra under operation Op when it is closed under Op.

The S leg of Birkhoff's HSP closure (#718 Slice 3), completing the triple with IsQuotientAlgebra (H, above in this partition) and IsProductAlgebra (P, also above). Mirrors :cartesian's IsCongruence shape (Slice 0): structural shape (the IsSubobject witness + Op signature gate) plus opt-in closure trait.

Universal-algebra references: Burris-Sankappanavar §II.5 (subalgebras) + §II.10 (the HSP closure operators).

Template Parameters

S	The candidate subalgebra (a Subobject of `A`).
A	The ambient algebra carrier.
Op	The binary operation `V` `×` `V` `→` `V` that `S` must be closed under.