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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.
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.
1. Internal direct sum: if E and F are abelian subgroups of an abelian group V, notation = means that V is the direct sum of E and F; that is, every element of V can be written in a unique way as the sum of an element of E and an element of F.
A PDF file is organized using ASCII characters, except for certain elements that may have binary content. The file starts with a header containing a magic number (as a readable string) and the version of the format, for example %PDF-1.7. The format is a subset of a COS ("Carousel" Object Structure) format. [23]
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 ...
Sum, an old name for the Finns in East Slavic languages, derived from the word Suomi, "Finland" Soum (currency) (also spelled "sum"), a unit of currency used in some Turkic-speaking countries of Central Asia; SUM (interbank network), an interbank network in 42 U.S. states; SUM, the ISO 639-3 code for the Sumo language; Cen (surname), sometimes ...
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Pronounced "A-star". A graph traversal and pathfinding algorithm which is used in many fields of computer science due to its completeness, optimality, and optimal efficiency. abductive logic programming (ALP) A high-level knowledge-representation framework that can be used to solve problems declaratively based on abductive reasoning. It extends normal logic programming by allowing some ...