
Re: function arity > 2
Posted:
Jan 16, 2013 1:51 AM


On Jan 15, 11:44 pm, JohnF <j...@please.see.sig.for.email.com> wrote: > As in universal algebra, where introductory discussions > typically suggest arity>2 not often used. Are there any > functions f:N^3>N (domain integers) that can't be > decomposed into some g,h:N^2>N, where f(i,j,k)=g(i,h(j,k))? > If so (i.e., if arity>2 needed), got an example? If not, > got a proof? And, if not for integers, is there any domain D > where f:D^3>D can't be decomposed like that (example or > proof again appreciated)?
Hmm. Your equation "f(i,j,k) = g(i,h(j,k))" seems kind of special, and it's not clear to me that it covers all ways of expressing a ternary operation in terms of binary operations. Anyway:
If D is an infinite domain, assuming the axiom of choice, there is an injection h:DxD>D, i.e., a pairing function. Given any function f:D^3>D, we can then define g(i,h(j,k)) = f(i,j,k).
If D = {0,1}, the function f:D^3>D such that f(0,j,k) = j, f(1,j,k) = k, can not be expressed in your form f(i,j,k) = g(i,h(j,k)).
If D is an nelement set, 2 < n < infinity, we can generalize the example for n = 2, or we can just use a counting argument: the number of ternary operations on D is n^(n^3), whereas the number of ordered pairs of binary operations is only n^(2n^2).

