## A further glance at classifiable 1-ary functions.(English)Zbl 0808.03016

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.

### MSC:

 03C45 Classification theory, stability, and related concepts in model theory
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### References:

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