## Notre Dame Journal of Formal Logic

### More on Generic Dimension Groups

Philip Scowcroft

#### Abstract

While finitely generic (f.g.) dimension groups are known to admit no proper self-embeddings, these groups also have no automorphisms other than scalar multiplications, and every countable infinitely generic (i.g.) dimension group admits proper self-embeddings and has automorphisms other than scalar multiplications. The finite-forcing companion of the theory of dimension groups is recursively isomorphic to first-order arithmetic, the infinite-forcing companion of the theory of dimension groups is recursively isomorphic to second-order arithmetic, and the first-order theory of existentially closed (e.c.) dimension groups is a complete $\Pi_{1}^{1}$-set. While many special properties of f.g. dimension groups may be realized in recursive e.c. dimension groups, and many special properties of i.g. dimension groups may be realized in hyperarithmetic e.c. dimension groups, no f.g. dimension group is arithmetic and no i.g. dimension group is analytical. Yet there is an f.g. dimension group recursive in first-order arithmetic, and (modulo a set-theoretic hypothesis) there is an i.g. dimension group recursive in second-order arithmetic.

#### Article information

Source
Notre Dame J. Formal Logic, Volume 56, Number 4 (2015), 511-553.

Dates
Received: 15 March 2012
Accepted: 28 June 2013
First available in Project Euclid: 30 September 2015

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1443620505

Digital Object Identifier
doi:10.1215/00294527-3153570

Mathematical Reviews number (MathSciNet)
MR3403090

Zentralblatt MATH identifier
1372.03081

#### Citation

Scowcroft, Philip. More on Generic Dimension Groups. Notre Dame J. Formal Logic 56 (2015), no. 4, 511--553. doi:10.1215/00294527-3153570. https://projecteuclid.org/euclid.ndjfl/1443620505

#### References

• [1] Addison, J. W., “Some consequences of the axiom of constructibility,” Fundamenta Mathematicae, vol. 46 (1959), pp. 337–57.
• [2] Ben Yaacov, I., A. Berenstein, C. W. Henson, and A. Usvyatsov, “Model theory for metric structures,” pp. 315–427 in Model Theory With Applications to Algebra and Analysis, Vol. 2, edited by Z. Chatzidakis, D. Macpherson, A. Pillay, and A. Wilkie, vol. 350 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 2008.
• [3] Elliott, G. A., “On the classification of inductive limits of sequences of semisimple finite-dimensional algebras,” Journal of Algebra, vol. 38 (1976), pp. 29–44.
• [4] Glass, A. M. W., and K. R. Pierce, “Equations and inequations in lattice-ordered groups,” pp. 141–71 in Ordered Groups (Boise, Idaho, 1978), edited by J. Smith, G. Kenny, and R. Ball, vol. 62 of Lecture Notes in Pure and Applied Mathematics, Dekker, New York, 1980.
• [5] Glass, A. M. W., and K. R. Pierce, “Existentially complete lattice-ordered groups,” Israel Journal of Mathematics, vol. 36 (1980), pp. 257–72.
• [6] Hirschfeld, J., and W. H. Wheeler, Forcing, Arithmetic, Division Rings, vol. 454 of Lecture Notes in Mathematics, Springer, Berlin, 1975.
• [7] Hodges, W., Building Models by Games, vol. 2 of London Mathematical Society Student Texts, Cambridge University Press, Cambridge, 1985.
• [8] Hodges, W., Model Theory, vol. 42 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1993.
• [9] Macintyre, A., “A note on axioms for infinite-generic structures,” Journal of the London Mathematical Society Second Series, vol. 9 (1974/75), pp. 581–84.
• [10] Robinson, J., “Definability and decision problems in arithmetic,” Journal of Symbolic Logic, vol. 14 (1949), pp. 98–114.
• [11] Rogers, H., Jr., Theory of Recursive Functions and Effective Computability, McGraw-Hill, New York, 1967.
• [12] Saracino, D., and C. Wood, “Finitely generic abelian lattice-ordered groups,” Transactions of the American Mathematical Society, vol. 277 (1983), pp. 113–23.
• [13] Scowcroft, P., “Some model-theoretic correspondences between dimension groups and AF algebras,” Annals of Pure and Applied Logic, vol. 162 (2011), pp. 755–85.
• [14] Scowcroft, P., “Existentially closed dimension groups,” Transactions of the American Mathematical Society, vol. 364 (2012), pp. 1933–74.
• [15] Weispfenning, V., “Model theory of abelian $l$-groups,” pp. 41–79 in Lattice-Ordered Groups: Advances and Techniques, edited by A. M. W. Glass and W. C. Holland, vol. 48 of Mathematics and Applications, Kluwer, Dordrecht, 1989.