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