Algebraically closed and existentially closed Abelian lattice-ordered groups

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.