P(n) = n-1. S(n) = n+1.
[P,S]n = PSn – SPn = 0
Here we assume Pn is defined so this is true. For n=0 in natural numbers, this is a problem. Note this is one way to lead to negative numbers. However, negative numbers mean we generalize natural numbers to ordered pairs of a sign and a natural number, (+,n) and (-,n) and define new functions P and S on this data.
A handy name for such ordered pairs is a sign and a size. This applies to fractions and reals as well.
Here P and S are unary operators.
Project: Define a commutator for binary operators, e.g. the commutator for addition and multiplication. What type of test data is needed?