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 | References | Similar Articles | Additional Information
Abstract: A partially ordered Abelian group is algebraically (existentially) closed in a class
of such structures just in case any finite system of weak inequalities (and negations of weak inequalities), defined over
, is solvable in
if solvable in some
in
. 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
(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
(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.
(1968), 125-132. MR 0227004 (37:2589)
- 7.
H. J. Keisler, Logic with the quantifier ``there exist uncountably many,'' Ann. Math. Logic
(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
(2011), 755-785.
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Additional Information
Philip Scowcroft
Affiliation:
Department of Mathematics and Computer Science, Wesleyan University, Middletown, Connecticut 06459
Email:
pscowcroft@wesleyan.edu
DOI:
https://doi.org/10.1090/S0002-9947-2011-05382-1
Received by editor(s):
January 2, 2010
Received by editor(s) in revised form:
January 13, 2010, and May 23, 2010
Published electronically:
November 17, 2011
Article copyright:
© Copyright 2011
American Mathematical Society