Concepts inRepresentation systems for data exchange
Data exchange
Data exchange is the process of taking data structured under a source schema and actually transforming it into data structured under a target schema, so that the target data is an accurate representation of the source data. Data exchange is similar to the related concept of data integration except that data is actually restructured (with possible loss of content) in data exchange. There may be no way to transform an instance given all of our constraints.
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Conjunctive query
In database theory, a conjunctive query is a restricted form of first-order queries. A large part of queries issued on relational databases can be written as conjunctive queries, and large parts of other first-order queries can be written as conjunctive queries. Conjunctive queries also have a number of desirable theoretical properties that larger classes of queries (e.g. , the relational algebra queries) do not share.
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Homomorphism
In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures. The word homomorphism comes from the ancient Greek language: ¿¿¿¿ (homos) meaning "same" and ¿¿¿¿¿ (morphe) meaning "shape". Isomorphisms, automorphisms, and endomorphisms are all types of homomorphism.
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Function composition
In mathematics, function composition is the application of one function to the results of another. For instance, the functions f: X ¿ Y and g: Y ¿ Z can be composed by computing the output of g when it has an argument of f(x) instead of x. Intuitively, if z is a function g of y and y is a function f of x, then z is a function of x. Thus one obtains a composite function g ¿ f: X ¿ Z defined by (g ¿ f)(x) = g(f) for all x in X.
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Closure (mathematics)
A set has closure under an operation if performance of that operation on members of the set always produces a member of the same set. For example, the real numbers are closed under subtraction, but the natural numbers are not: 3 and 8 are both natural numbers, but the result of 3 ¿ 8 is not a natural number. Another example is the set containing only the number zero, which is a closed set under multiplication.
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Composability
Composability is a system design principle that deals with the inter-relationships of components. A highly composable system provides recombinant components that can be selected and assembled in various combinations to satisfy specific user requirements.
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Algebraically closed field
In abstract algebra, an algebraically closed field F contains a root for every non-constant polynomial in F[x x], the ring of polynomials in the variable x with coefficients in F.
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Isomorphism
In abstract algebra, an isomorphism is a bijective homomorphism. Two mathematical structures are said to be isomorphic if there is an isomorphism between them. In category theory, an isomorphism is a morphism f: X ¿ Y in a category for which there exists an "inverse" f: Y ¿ X, with the property that both ff = idX and f f = idY.
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