dedekind::algebra::AlgebraAsProduct Concept Reference

Set-level sibling of HasCarrier: an algebra is a product of (set object, operations on the underlying type). More...

Concept definition

template<typename A, typename Set, typename... Ops>
concept dedekind::algebra::AlgebraAsProduct = 
    dedekind::category::IsSubobject<Set, typename Set::Domain> &&
    HasCarrier<A, typename Set::Domain, Ops...>

Detailed Description

Set-level sibling of HasCarrier: an algebra is a product of (set object, operations on the underlying type).

Where element-level HasCarrier<Algebra, Carrier, Ops...> reads the carrier as a TYPE (e.g. int, bool, unsigned), this set-level concept reads the carrier as a SET OBJECT — something with both an underlying species and a characteristic morphism χ. In universal- algebra vocabulary (Burris–Sankappanavar §I.1): every algebra (A, F) is a pair of (carrier set A, operations F); element-level HasCarrier names the (A_type, F) reading, this set-level concept names the full (A_set, F) reading where A_set is the categorical set object carrying both the underlying type and the membership predicate.

The two concepts are siblings, not replacements:

• HasCarrier: the C++-idiomatic form ("algebra over @c int")
• AlgebraAsProduct: the textbook-idiomatic form ("algebra over a set")

Composition (read as &&-as-intersection):

AlgebraAsProduct<A, Set, Ops...>
  := IsSubobject<Set, Set::Domain>        (Set is a set object)
   ∧ HasCarrier<A, Set::Domain, Ops...>   (A has Set::Domain as
                                            carrier with these Ops)

Sollbruchstelle (#573)

Names the (Set, Operations) product reading of an algebra explicitly, complementing slice 2's SetAsProduct<Set, Underlying, Classifier> in :category:concrete. Together they encode the nested-product structure algebra → set → underlying as typed obligations rather than naming conventions. The pilot witness on a real named carrier (Rational) ships in #573 slice 4 (#646).

Template Parameters

A	The algebra (whole).
Set	The underlying set object (an IsSubobject; the left part of the (Set, Operations) pair).
Ops	The operations on Set::Domain (the right part).

See also

HasCarrier above (element-level sibling, this partition).

SetAsProduct (set-level sibling, :category:concrete; #644 slice 2).

Burris & Sankappanavar 1981 §I.1 (algebra (A, F) pair).