New axioms for the lattice-ordered groups existentially closed in $\mathbf W^+$

Let $\bf {W}^+$ be the class of nonzero Archimedean lattice-ordered groups with distinguished strong order unit, viewed as structures for the first-order language $\{ +, -, \wedge , \vee , 0, 1 \}$. This paper gives new axioms for the lattice-ordered groups existentially closed in $\bf {W}^+$ and uses them to show that $(C(X),1_X)$ is existentially closed in $\bf {W}^+$ if and only if $X$ is nonempty, pseudocompact, an almost-$P$-space, and a strongly zero-dimensional $F$-space with no isolated points.