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) = βf(a) This equation is depicted by the commutative diagram shown in the figure.
ϕ : X → Y
α : A → X
β : B → Y
∅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.