A singleton generator
THEOREM: Let X be any infinite set, and let g1, g2, ...
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 gi; that is, every
gi can be obtained by nesting sufficiently many
instances of +, e.g., g1(x,y,z) could be
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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