How to produce random functions $g:N\times N\rightarrow N$ where $N=\{ 0,1,\ldots ,n \}$ and $$g\Big( g(a,b)\; ,c\Big)=g\Big(a,\; g(b,c)\Big)\quad \forall a,b,c\in N \qquad \text{[Constraint of associativity.]}$$
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Have you done anything to characterize this class of functions? – A rural reader Jul 01 '21 at 19:24
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A few examples are $g(x, y) = x + y$, $g(x, y) = xy$, $g(x, y) = \max(x, y)$, $g(x, y) = \min(x, y)$, etc. – A rural reader Jul 01 '21 at 20:06