Existentially closed dimension groups

Scowcroft, Philip

doi:10.1090/S0002-9947-2011-05382-1

Existentially closed dimension groups

Author: Philip Scowcroft
Journal: Trans. Amer. Math. Soc. 364 (2012), 1933-1974
MSC (2010): Primary 03C60, 06F20; Secondary 03C25
DOI: https://doi.org/10.1090/S0002-9947-2011-05382-1
Published electronically: November 17, 2011
MathSciNet review: 2869195
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Abstract: A partially ordered Abelian group $\mathcal {M}$ is algebraically (existentially) closed in a class $\mathcal {C}\ni \mathcal {M}$ of such structures just in case any finite system of weak inequalities (and negations of weak inequalities), defined over $\mathcal {M}$ , is solvable in $\mathcal {M}$ if solvable in some $\mathcal {N}\supseteq \mathcal {M}$ in $\mathcal {C}$ . After characterizing existentially closed dimension groups this paper derives amalgamation properties for dimension groups, dimension groups with order unit, and simple dimension groups. By determining the quantifier-free types that may be isolated by existential formulas the paper produces many pairwise nonembeddable countable finitely generic dimension groups. The paper also finds several elementary properties distinguishing finitely generic dimension groups among existentially closed dimension groups. The paper finally embeds nontrivial dimension groups functorially into existentially closed dimension groups.

1. I. Ben Yaacov, A. Berenstein, C. W. Henson, and A. Usvyatsov, Model theory for metric structures. In Z. Chatzidakis, D. Macpherson, A. Pillay, and A. Wilkie, eds., Model Theory with Applications to Algebra and Analysis, Volume 2, London Math. Soc. Lecture Note Ser. No. 350, Cambridge University Press, Cambridge, 2008, pp. 315-427. MR 2436146 (2009j:03061)
2. K. B. Bruce, Model-theoretic forcing in logic with a generalized quantifier, Ann. Math. Logic $\mathbf {13}$ (1978), 225-265. MR 0491860 (80c:03033)
3. G. Elliott, On the classification of inductive limits of sequences of semisimple finite-dimensional algebras, J. Algebra $\mathbf {38}$ (1976), 29-44. MR 0397420 (53:1279)
4. L. Fuchs, Riesz vector spaces and Riesz algebras, Queen's Papers in Pure and Appl. Math. No. 1, Queen's University, Kingston, Ontario, 1966. MR 0203436 (34:3288)
5. W. Hodges, Building models by games, London Math. Soc. Stud. Texts No. 2, Cambridge University Press, Cambridge, 1985. MR 0812274 (87h:03045)
6. B. Jónsson and P. Olin, Almost direct products and saturation, Compos. Math. $\mathbf {20}$ (1968), 125-132. MR 0227004 (37:2589)
7. H. J. Keisler, Logic with the quantifier ``there exist uncountably many,'' Ann. Math. Logic $\mathbf {1}$ (1970), 1-93. MR 0263616 (41:8217)
8. P. Scowcroft, A representation of convex semilinear sets, Algebra Universalis 62 (2009), 289-327. MR 2661381 (2011h:52007)
9. P. Scowcroft, Some model-theoretic correspondences between dimension groups and AF algebras, Ann. Pure Appl. Logic $\mathbf {162}$ (2011), 755-785.