## Quick Reference

A method for displaying equations between functions. For example, suppose that there is a function ∅ of the form *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 ∅ on the code sets *A* and *B* if, for all *a* ∈ *A*, the following equation holds: *∅α*(*a*) = β*f*(*a*) This equation is depicted by the commutative diagram shown in the figure.

*ϕ* : *X* → *Y*

*α* : *A* → *X*

β : *B* → *Y*

*∅α*(*a*) = β*f*(*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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