User blog comment:Alejandro Magno/ExE array of/@comment-2033667-20141021222936/@comment-4224897-20141021235145

A definition is a set of rules that tells you how to evaluate any expression in the notation, so that no gaps have to be filled with intution. A simple example is the definition of up-arrows:

Rule 1: x↑y = x^y

Rule 2: x↑ay = x↑a(x↑a(y-1)) if y isn't 0

Rule 3: x↑a0 = 1

(↑a means ↑↑↑...↑↑↑ with a ↑s)

The definitions need not be easy to understand (e.g. the definition of Extended Cascading-E or Hollom's hyperfactorials), which is why examples help a lot with that. However, you should avoid relying solely on examples for defining a notation.