Which are all the associative binary operations on natural numbers ? Certain results in this regard can be found in arxiv:math/0508215. It appears that such associative operations cannot grow too fast. Or perhaps, there are also those which have to grow very fast. It is easy to prove that there are infinitely many such associative operations. And in fact, uncountably many, as shown in the mentioned paper. Amusingly, under rather simple natural conditions, the usual addition and multiplication are the only associative binary operations on natural numbers, as wes proved back in 1981, and cited in the mentioned paper. This fact is the motivation for the above question, namely, what happens if we do not ask any conditions on such associative binary operations ? The whole story arose from the observation that certain minimal algebraic and topological axioms determine uniquely the sets on which they hold. An example is the theorem of Pontriaghin proved in the 1930s that the only commutative and algebraically closed field which is not discrete and it is complete is that of complex numbers. And then, one can turn the issue around and ask whether there are simplest infinite sets which determine uniquely some of the algebraic, topological, or other structures on them ? Well, it turns out the the set of natural integers just about determines uniquely addition and multiplication.
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