## On superstable expansions of free abelian groups.(English)Zbl 1455.03042

Summary: We prove that $$(\mathbb{Z},+,0)$$ has no proper superstable expansions of finite Lascar rank. Nevertheless, this structure equipped with a predicate defining powers of a given natural number is superstable of Lascar rank $$\omega$$. Additionally, our methods yield other superstable expansions such as $$(\mathbb{Z},+,0)$$ equipped with the set of factorial elements.

### MSC:

 03C45 Classification theory, stability, and related concepts in model theory 03C60 Model-theoretic algebra

### Keywords:

superstability; free abelian groups
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### References:

 [1] Baldwin, J., and M. Benedikt, “Stability theory, permutations of indiscernibles, and embedded finite models,” Transactions of the American Mathematical Society, vol. 352 (2000), pp. 4937-69. · Zbl 0960.03027 [2] Casanovas, E., and M. Ziegler, “Stable theories with a new predicate,” Journal of Symbolic Logic, vol. 66 (2001), pp. 1127-40. · Zbl 1002.03023 [3] Chernikov, A., and P. Simon, “Externally definable sets and dependent pairs, II,” Transactions of the American Mathematical Society, vol. 367 (2015), pp. 5217-35. · Zbl 1388.03035 [4] Marker, D., “A strongly minimal expansion of $$(ω,s)$$,” Journal of Symbolic Logic, vol. 52 (1987), pp. 205-7. · Zbl 0647.03029 [5] Pillay, A., Geometric Stability Theory, vol. 32 of Oxford Logic Guides, Oxford University Press, New York, 1996. · Zbl 0871.03023 [6] Pillay, A., and C. Steinhorn, “Discrete o-minimal structures,” Annals of Pure and Applied Logic, vol. 34 (1987), pp. 275-89. · Zbl 0623.03038 [7] Poizat, B., “Supergénérix,” Journal of Algebra, vol. 404 (2014), pp. 240-70. [8] Sela, Z., “Diophantine geometry over groups, VIII: Stability,” Annals of Mathematics (2), vol. 177 (2013), 787-868. · Zbl 1285.20042 [9] Sklinos, R., “Some model theory of the free group,” Ph.D. dissertation, University of Leeds, Leeds, United Kingdom, 2011. · Zbl 1213.03047 [10] Wagner, F. O., Stable Groups, vol. 240 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1997. · Zbl 0897.03037 [11] Wagner, F. O., “Some remarks on one-basedness,” Journal of Symbolic Logic, vol. 69 (2004), pp. 34-38. · Zbl 1073.03018
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