In the context of continuous logic, this paper axiomatizes both the class $$\mathcal {C}$$ C of lattice-ordered groups isomorphic to C(X) for X compact and the subclass $$\mathcal {C}^+$$ C+ of structures existentially closed in $$\mathcal {C}$$ C ; shows that the theory of $$\mathcal {C}^+$$ C+ is $$\aleph _0$$ ℵ0 -categorical and admits elimination of quantifiers; establishes a Nullstellensatz for $$\mathcal {C}$$ C and $$\mathcal {C}^+$$ C+ ; shows that $$C(X)\in \mathcal {C}$$ C(X)∈C has a prime-model extension in $$\mathcal {C}^+$$ C+ just in case X is Boolean; and proves that in a sense relevant to continuous logic, positive formulas admit in $$\mathcal {C}^+$$ C+ elimination of quantifiers to positive formulas.