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 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.

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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