# commutative diagram

**commutative diagram** A method for displaying equations between functions. For example, suppose that there is a function φ *φ* : *X* → *Y*

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 :* *A* → *X* and β : *B* → *Y*

and a function *f* : *A* → *B* is defined to be a representation or function for φ on the code sets *A* and *B* if, for all *a*∈*A*, the following equation holds: *φα(**a*) = β*f*(*a*)

This equation is depicted by the commutative diagram shown in the figure.

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.

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**commutative diagram**