A further glance at classifiable 1-ary functions (Q1332500)

The aim of this paper is to study Shelah's classification theory for \(k\)- tuples of 1-ary functions \(f_ 0,\dots, f_{k-1}\) when \(k\geq 2\). It is easy to see that if \(f_ j\) is 1-1 for all \(j<k\), or, more generally, if the power of \(f^{-1}_ j(a)\) is uniformly bounded for all \(j< k\) and for all \(a\), then the complete theory of \(f_ 0,\dots,f_{k-1}\) is classifiable. But here it is shown that if one weakens the previous conditions by considering pairs \((f_ 0,f_ 1)\) of 1-ary functions such that \(f_ 0\) is 1-1 and, for all terms \(s\) and \(t\) of the language with \(s\neq t\), the sentence \(\forall\vec v(s(\vec v)\neq t(\vec v))\) holds, then there is a function \(F\) mapping any graph \((X,R)\) into such a pair \((f_ 0,f_ 1)\), preserving and reflecting isomorphism, elementary equivalence and classifiability of the corresponding theories. So, in these cases, the theory of \((f_ 0,f_ 1)\) is very far from being classifiable.