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It is well-known that the theory of the structure $(\mathbb{N},<)$ is not stable, but is NIP and has quantifier elimination in the language $L=\{<,0,S,S^{-1}\}$ where $S,S^{-1}$ are function symbols represented as the successor and predecessor functions, respectively.

Consider now an irrational number $\alpha\in (0,1)$ and the function $f:\mathbb{N}\to\mathbb{N}$ interpreted as $f(x)=\lfloor \alpha\cdot x\rfloor$ (the integer part of $\alpha\cdot x$).

Questions:

  1. Is there any natural language on which the theory $\operatorname{Th}(\mathbb{N},<,f)$ has quantifier elimination?
  2. Is the theory $\operatorname{Th}(\mathbb{N},<,f)$ still NIP?
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    Just as a point of interest, generally, theories are not stable, NIP etc. The same sort of comments hold for quantifier elimination: the theory of DLO with endpoints admits QE for example. – 2017-01-18
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    @dav11 Are you asserting that a "random" complete theory (interpreted in a nontechnical sense) will have all "bad" properties (e.g. have OP, IP, SOP, not have QE, etc.)? Also, you probably mean to say that DLO with endpoints does *not* admit QE. – 2017-01-18
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    (I'm not saying I disagree with that assertion, I just wasn't sure I understood the meaning of your comment) – 2017-01-19
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    I think dav11 meant that NIP and QE are properties of theories, but I was asking te question about a structure – 2017-01-19
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    @AlexKruckman: No, not at all. I just meant that stable, NIP etc are usually properties of theories and not of a particular structure. And you are correct; I meant to say DLO without! not with! – 2017-01-19
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    Oh, now I understand! In my experience, it is fairly common to call a structure stable (for example) if it's complete theory is stable. This is especially common in the case of o-minimality, where you frequently hear "let $M$ be an o-minimal structure". – 2017-01-19
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    @AlexKruckman: Hmmm, that is true. I guess I'm just used to referring to theories. – 2017-01-19
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    @Wore: Where did you find this question? – 2017-01-20
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    @KyleGannon It is a variation of a problem I tried to work out for my thesis, about classes of finite linear orders with additional structure. – 2017-01-20

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