In this article I will discuss array indexing, operators, and composition in depth. If you work through this article you should end up with a very deep understanding of array indexing and the deep interpretation available when we realize indexing is an instance of function composition (or an example of permutation groups or semigroups: some *very deep* yet accessible pure mathematics).

A permutation of indices

In this article I will be working hard to convince you a very fundamental true statement is in fact true: array indexing is associative; and to simultaneously convince you that you should still consider this amazing (as it is a very strong claim with very many consequences). Array indexing respecting associative transformations should not be a-priori intuitive to the general programmer, as array indexing code is rarely re-factored or transformed, so programmers tend to have little experience with the effect. Consider this article an exercise to build the experience to make this statement a posteriori obvious, and hence something you are more comfortable using and relying on.

`R`

‘s array indexing notation is really powerful, so we will use it for our examples. This is going to be long (because I am trying to slow the exposition down enough to see all the steps and relations) and hard to follow without working examples (say with `R`

), and working through the logic with pencil and a printout (math is not a spectator sport). I can’t keep all the steps in my head without paper, so I don’t really expect readers to keep all the steps in their heads without paper (though I have tried to organize the flow of this article and signal intent often enough to make this readable). Continue reading On indexing operators and composition