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Algebra universalis

, Volume 75, Issue 3, pp 257–300 | Cite as

Algebraically closed and existentially closed Abelian lattice-ordered groups

  • Philip ScowcroftEmail author
Article

Abstract

If \({\mathcal{G}}\) is an Abelian lattice-ordered (l-) group, then \({\mathcal{G}}\) is algebraically (existentially) closed just in case every finite system of l-group equations (equations and inequations), involving elements of \({\mathcal{G}}\), that is solvable in some Abelian l-group extending \({\mathcal{G}}\) is solvable already in \({\mathcal{G}}\). This paper establishes two systems of axioms for algebraically (existentially) closed Abelian l-groups, one more convenient for modeltheoretic applications and the other, discovered by Weispfenning, more convenient for algebraic applications. Among the model-theoretic applications are quantifierelimination results for various kinds of existential formulas, a new proof of the amalgamation property for Abelian l-groups, Nullstellensätze in Abelian l-groups, and the display of continuum-many elementary-equivalence classes of existentially closed Archimedean l-groups. The algebraic applications include demonstrations that the class of algebraically closed Abelian l-groups is a torsion class closed under arbitrary products, that the class of l-ideals of existentially closed Abelian l-groups is a radical class closed under binary products, and that various classes of existentially closed Abelian l-groups are closed under bounded Boolean products.

Key words and phrases

lattice-ordered group algebraically closed existentially closed finitely generic infinitely generic 

2010 Mathematics Subject Classification

Primary: 03C60 Secondary: 06F20 03C25 

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Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceWesleyan UniversityMiddletownUSA

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