View source for Example:Bijective undo/redo operations

Many user interfaces can be thought of as a collection of [[function]]s that transform a document into another document.  For example, let <math>A</math> be the set of all plain text files.  Pressing the 'x' key in a text editor causes an 'x' to be inserted; you can think of this as applying a function <math>f : A [[\to]] A</math> that takes the document without the 'x' and outputs the document with the 'x'.

If you want to provide an undo/redo capability, then the [[function]]s should be [[bijection]]s.  If they are, then their [[two-sided inverse]]s give the undo operations.

Sometimes you can make a [[function]] into a [[bijection]] by adding an "undo log": by expanding the set <math>A</math> you can keep enough history to implement an undo function.