Is there a theorem or axiom which shows that permutations of step sequences through a lattice graph result in the same destination?

Question

I have been searching for a theorem, lemma, or even an axiom which shows that the permutations of a step sequence in Taxicab Geometry result in the same destination in such a lattice graph.

To clarify, let us take N, S, E, and W to represent North, South, East, and West, respectively. Then, as a simple example, the step sequences, written as a string/word, all result in the same destination: (NEE), (ENE), and (EEN). In Cartesian Coordinates, this shared destination would, of course, be (2, 1).

So far, all I have been able to find are theorems about the number of such lattice paths, and, subsequently the Catalan Numbers, Delannoy Numbers, and the like.

Is there such a theorem which shows this fact? or is it an axiom? is it a consequence of the commutativity property of these strings/words?

I am currently researching a topic which relies on this particular truth, and would thus like to make the appropriate references.

Thanks!

Answer 1

$N$ steps increment the $y$ coordinate.

$E$ steps increment the $x$ coordinate.

So, sequences of $N$ and $E$ with same number of each go to the same point, regardless of order.

If you have sequences of $N$, $S$, $E$, $W$, then $N$ and $S$ cancel and $E$ and $W$ cancel. So, sequences with the same $N-S$ and $E-W$ go to the same point, regardless of order.

Answer 2

Hint: You could prove that the strings in your language represent an abelian group.

Answer 3

I worked out the answer to my own question right after I posted it. For my purposes, the commutativity of vector addition is the key idea. It was just a case of "losing sight of the forest for the trees." The reference I am looking for might very well be to a textbook on basic Linear Algebra.

Thank you for your insights, though. They might nevertheless prove useful.

As a follow up question, if anything, should I refer to the commutativity of vector addition as an axiom? or a fundamental theorem? would this depend on how we construct the vector space?

Thanks again.