Unique representations of positive integers?

Question

I was thinking about unique representations of the positive integers by a subset of the positive integers.

For the representation every element of the subset of positive integers may be used AT MOST ONCE.

So we consider unique representations of the positive integers by some operator and some subset of the positive integers where every element of the subset may be used at most once.

For the (operator) sum this subset is the powers of 2. For the (operator) product this subset is the primes and the prime powers.

These are the only 2 solutions I know about.

What OTHER unique representations exist for the positive integers ?

To avoid confusion and mistakes ;

1) I know every positive integer is the sum of 4 squares THAT IS NOT UNIQUE HOWEVER AND THUS NO (NEW) VALID ANSWER.

2) Making a bijection from a subset of the integers to the primes , prime powers or powers of 2 is ALSO NO (NEW) VALID ANSWER.

3) EVERY operator (of the answer) between any elements of the set should be a positive integer !

Im not sure there are such other unique representations , but I cannot (dis)prove that either.

My apologies for the late edit , I do not have alot of time.

Maybe this is better as a comment but Im confused why the tag " representation " has been removed.

edit :

the factorial number system shows how other unique representations can be given for the sum operator.

So this reduces the question for operators that are not the simple sum.

Answer 1

You can rephrase your question as follows: What are the $\mathbb F_2$ vector space structures on the natural numbers?

A unique representation via a fixed subset and an operation is nothing else then the unique expression using a basis. In order to answer the question: Proof that every infinite set K has the same cardinality as the set of its finite subsets. The latter clearly has a $\mathbb F_2$ vector space structure by the symmetric difference.