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In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. The direct sum of modules is the smallest module which contains the given modules as submodules with no "unnecessary" constraints, making it an example of a coproduct. Contrast with the direct product, which is the dual notion.
The direct sum is also commutative up to isomorphism, i.e. for any algebraic structures and of the same kind. The direct sum of finitely many abelian groups, vector spaces, or modules is canonically isomorphic to the corresponding direct product. This is false, however, for some algebraic objects, like nonabelian groups.
The group operation in the external direct sum is pointwise multiplication, as in the usual direct product. This subset does indeed form a group, and for a finite set of groups {H i} the external direct sum is equal to the direct product. If G = ΣH i, then G is isomorphic to Σ E {H i}. Thus, in a sense, the direct sum is an "internal ...
In particular, the concept of universal property allows a simple proof that all constructions of real numbers are equivalent: it suffices to prove that they satisfy the same universal property. Technically, a universal property is defined in terms of categories and functors by means of a universal morphism (see § Formal definition, below).
Universal property of the product Whether a product exists may depend on C {\displaystyle C} or on X 1 {\displaystyle X_{1}} and X 2 . {\displaystyle X_{2}.} If it does exist, it is unique up to canonical isomorphism , because of the universal property, so one may speak of the product.
The direct product G × H can be characterized by the following universal property. Let π G: G × H → G and π H: G × H → H be the projection homomorphisms. Then for any group P and any homomorphisms ƒ G: P → G and ƒ H: P → H, there exists a unique homomorphism ƒ: P → G × H making the following diagram commute:
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This pattern holds for any variety in the sense of universal algebra. The coproduct in the category of Banach spaces with short maps is the l 1 sum, which cannot be so easily conceptualized as an "almost disjoint" sum, but does have a unit ball almost-disjointly generated by the unit ball is the cofactors. [1]