# A singleton generator

THEOREM: Let X be any infinite set, and let g_{1}, g_{2}, ...
countably many
operations on X (i.e., functions from some finite power of X into X).
Then there is a single binary operation "+" on X such that the
clone generated by "+" contains each g_{i}; that is, every
g_{i} can be obtained by nesting sufficiently many
instances of +, e.g., g_{1}(x,y,z) could be
z+((y+x)+x).
A pdf version is available.

The proof is easy, and perhaps quite old
and well-known. (Does anybody know a reference?
Please write me.)

The same theorem is true for finite sets, with a different proof.
(Donald Webb, 1935)

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