# Algebraically closed and existentially closed Abelian lattice-ordered groups

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## 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## Preview

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