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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:

[1] J. Baldwin , Fundamentals of Stability Theory, Perspectives in Mathematical Logic , Springer , Berlin ( 1988 ). MR 918762 | Zbl 0685.03024 · Zbl 0685.03024
[2] J. Flum , First Order Logic and its Extensions , Lecture Notes in Mathematics , 499, Springer , Berlin ( 1975 ), pp. 248 - 310 . MR 401465 | Zbl 0342.02007 · Zbl 0342.02007
[3] A. Marcja - M. Prest - C. Toffalori , Classification theory for Abelian groups with an endomorphism , Arch. Math. Logic , 31 ( 1991 ), pp. 95 - 104 . MR 1139394 | Zbl 0723.03020 · Zbl 0723.03020 · doi:10.1007/BF01387762
[4] A. Mekler , Stability of nilpotent groups of class 2 and prime exponent , J. Symbolic Logic , 46 ( 1981 ), pp. 781 - 788 . MR 641491 | Zbl 0482.03014 · Zbl 0482.03014 · doi:10.2307/2273227
[5] C. Toffalori , Classification theory for a 1-ary function , Illinois J. Math. , 35 ( 1991 ), pp. 1 - 26 . Article | MR 1076663 | Zbl 0696.03018 · Zbl 0696.03018
[6] C. Toffalori , 1-ary functions and the f. c. p. , Illinois J. Math. , 35 ( 1991 ), pp. 434 - 450 . Article | MR 1103677 | Zbl 0716.03027 · Zbl 0716.03027
[7] C. Toffalori , The stability problem for 2-ary functions , Boll. Un. Mat. Ital . ( 7 ), 7-B ( 1993 ), pp. 187 - 200 . MR 1216715 | Zbl 0778.03008 · Zbl 0778.03008
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