# How many associative binary operations there are on a finite set?

I am studying Scott's book Group Theory. In the Exercise $1.1.17$ he asks us to show that if $S$ is a set and $|S|=n$, then there are $n^{\frac{n^{2}+n}{2}}$ commutative binary operations on $S$. But he doesn't talk about how many associative binary operations there are on a finite set.

Is there an answer to that question? I mean, how many associative binary operations there are on a finite set?

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Knowing how many associative binary operations there are on a set with $n$ elements would tell you how many non-isomorphic semigroup structures there are on a set with $n$ elements. As far as I know, there is no known closed formula; a recent (2010) thesis gives the number of non-isomorphic, non-anti-isomorphic semigroups with $9$ elements as 52,989,400,714,478. –  Arturo Magidin Feb 3 '12 at 21:38
–  Arturo Magidin Feb 3 '12 at 21:43
@ArturoMagidin: Thanks for the references. –  spohreis Feb 3 '12 at 22:58
@spohreis: On this site, Doug Spoonwood once asked whether the proportion of associative binary operations to total operations approached $0$ as $n$ gets large. I gave an answer that you can find here. But the estimate I gave is very weak, just enough to show that the limit is $0$. –  André Nicolas Feb 4 '12 at 0:51