commutative diagram

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A method for displaying equations between functions. For example, suppose that there is a function ∅ of the form ϕ : XY and what is needed is to represent or code the data in X and Y, and the function, by means of the data sets A and B, respectively. Functions α and β are chosen where α : AX and β : BY and a function f : AB is defined to be a representation or function for ∅ on the code sets A and B if, for all aA, the following equation holds: ∅α(a) = βf(a) This equation is depicted by the commutative diagram shown in the figure.

ϕ : XY

α : AX

β : BY

∅α(a) = βf(a)

Equations and commutative diagrams of this form play an important role in relating different levels of abstraction, and are used to formulate the correctness of data-type implementations, compilers, and machine architectures. As equations grow in complexity, commutative diagrams become essential. See also computable algebra.

Commutative diagram.

Subjects: Computing.

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