Mekler’s construction and generalized stability
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Mekler’s construction gives an interpretation of any structure in a finite relational language in a group (nilpotent of class 2 and exponent p > 2, but not finitely generated in general). Even though this construction is not a bi-interpretation, it is known to preserve some model-theoretic tameness properties of the original structure including stability and simplicity. We demonstrate that k-dependence of the theory is preserved, for all k ∈ N, and that NTP2 is preserved. We apply this result to obtain first examples of strictly k-dependent groups (with no additional structure).
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- H. Adler, An introduction to theories without the independence property, https://doi.org/www.logic.univie.ac.at/~adler/docs/nip.pdf
- A. Chernikov, D. Palacin and K. Takeuchi, On n-dependence, Notre Dame Journal of Formal Logic, to appear, arXiv:1411.0120.Google Scholar
- W. Hodges, Model Theory, Encyclopedia of Mathematics and its Applications, Vol. 42, Cambridge University Press, Cambridge, 1993.Google Scholar
- I. Kaplan and N. Ramsey, On Kim-independence, Journal of the European Mathematical Society, to appear, arXiv:1702.03894.Google Scholar
- S. Shelah, Classification Theory and the Number of Non-isomorphic Models, Studies in Logic and the Foundations of Mathematics, Vol. 92, North-Holland, Amsterdam, 1990.Google Scholar
- S. Shelah, Definable groups for dependent and 2-dependent theories, Sarajevo Journal of Mathematics 13(25) (2017), 3–25.Google Scholar
- F. O. Wagner, Simple Theories, Mathematics and its Applications, Vol. 503, Kluwer Academic, Dordrecht, 2000.Google Scholar