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Can there exist a ring structure on the natural numbers? So far, I have thought of one operation, $$a+b=\max\{a,b\} $$ but even in this case, the identity element depends on each natural number. Can such a structure exist? If not, can it be proven that the natural numbers can never admit a ring or group structure?

Θάνος Κ.
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  • What woul the opposite of an element be ? – Jean Marie Sep 16 '24 at 13:48
  • The operation you describe does not have additive inverses and gives rise to a semiring (see "max-plus algebra") where usual addition is the ring product – Ben Sep 16 '24 at 13:50
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    This isn't clear. Which properties of the natural numbers do you want the ring structure to preserve? – lulu Sep 16 '24 at 13:50
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    If all you want is $\mathbb N$ as a countable set, then sure. Fix a bijection with $\mathbb Z$ and pull back the ring structure – lulu Sep 16 '24 at 13:52
  • I am looking for operations that can create a ring structure on the natural numbers. – Θάνος Κ. Sep 16 '24 at 14:02
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    Then see lulu's commentr. If $f:\Bbb N\to\Bbb Z$ is any bijection, then $m\oplus n:=f^{-1}(f(m)+f(n))$ and $m\otimes n:=f^{-1}(f(m)\times f(n))$ define a ring structure on $\Bbb N$. (This is called "transport of structure.") – coiso Sep 16 '24 at 14:20
  • There is a relatively simple definition, if you don't require a multiplicative identity. (Most definitions of ring do, some don't.) Namely, $+$ is the bitwise "xor," and $\times$ is the bitwise "and" – Thomas Andrews Sep 16 '24 at 15:05

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