Is there a finite generating set for the clone of all operations on a finite set with at least 3 elements?

Question

Let $S$ be a finite set with at least $3$ elements, and let $O(S)$ be the set of all finitary operations on $S$. That is, it is the set of all unary, all binary, all ternary, etc operations collected in one big set. Does there exist a finite set of operations such that the clone generated by that set is $O(S)$?

Answer 1

The answer is yes, and in fact $O(S)$ can be generated by a single binary function. For example, in 1936 Webb proved that the function $$(x,y)\mapsto [\min\{x,y\}+1]_{\mbox{mod } \vert S\vert}$$ generates the whole clone $O(S)$. See also Webb's thesis and this hsmse question, as well as Graham 1967.

More generally, there is a wealth of information about such topics in Lau's book Function algebras on finite sets, although it can be a bit tricky to navigate. Note that your $O(S)$ is Lau's $P_{\vert S\vert}$.

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There is an important (if elementary) caveat here: since there are uncountably many clones on a finite set with $>2$ elements, it is not the case that "big = complicated" for clones: a subclone of a $1$-generated clone need not be finitely generated. Of course there is no reason to be surprised by this, but it's still worth mentioning.