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I want to find a function $L:\mathbb{N}\times\mathbb{N}\rightarrow\mathbb{N}$ satisfies: $$ L(a,b)=L(b,a) $$


$$ L(a,L(b,c))=L(L(a,b),c) $$

I tried several functions, but only two trivial solution works, i.e. $L(m,n)=m+n+C$ or $L(m,n)=Cmn$.

So I wonder are they the only solution? If not, please show me an example..

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up vote 4 down vote accepted

Consider $L(m,n) = mn + m+ n$.

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And they said studying for the Math GRE was a waste of time... – Isaac Solomon Nov 11 '12 at 9:28
In fact, I'm considering how to give a different ring structure on $\mathbb{N}$. Using your $L(m,n)$ as addition, is there a suitable multiplication?..Or, do you know an different ring structure on $N$? – hxhxhx88 Nov 11 '12 at 9:34
@hxhxhx88: $\mathbb{N}$ with the usual operations isn't a ring; and it also isn't a ring with $m \cdot n = L(m,n)$ as its addition (or multiplication). – Clive Newstead Nov 11 '12 at 9:35
He probably meant Z – Amr Nov 11 '12 at 9:36
Again let p:Z→Z be a bijection, then define A(m,n)=$p^{-1}(p(m)+p(n))$ and define $M(m,n)=p^{-1}(p(m)p(n))$. The structure (Z,A,M) is a ring – Amr Nov 11 '12 at 9:54

Another function L(a,b)=(ab)mod 2

let $p:Z→Z$ be a bijection, then define $A(m,n)=p^{−1}(p(m)+p(n))$ and define $M(m,n)=p^{−1}(p(m)p(n))$. The structure (Z,A,M) is a ring. In fact, by Letting p(n)=n+1 we find that $M(m,n)=p^{−1}(p(m)p(n))=mn+m+n$ and $A(m,n)=m+n+1$

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