A function is an object that you can create, manipulate, pass to, and return from functions (for some interesting examples of library functions that manipulate functions, see MappingPackage1XmpPage ). Yet, we often seem to use the term operation and function interchangeably in Axiom. What is the distinction?
First consider values and types associated with some variable in your workspace. You can make the declaration n : Integer, then assign an integer value. You then speak of the integer . However, note that the integer is not the name itself, but the value that you assign to .
Similarly, you can declare a variable in your workspace to have type Integer Integer, then assign , through a definition or an assignment of an anonymous function. You then speak of the function . However, the function is not , but the value that you assign to .
A function is a value, in fact, some machine code for doing something. Doing what? Well, performing some operation. Formally, an operation consists of the constituent parts of in your workspace, excluding the value; thus an operation has a name and a type. An operation is what domains and packages export. Thus Ring exports one operation ``+''. Every ring also exports this operation. Also, the author of every ring in the system is obliged under contract (see ugPackagesAbstract ) to provide an implementation for this operation.
This chapter is all about functions---how you create them interactively and how you apply them to meet your needs. In Chapter ugPackages you will learn how to create them for the Axiom library. Then in Chapter ugCategories , you will learn about categories and exported operations.