May 12
Brian Wynne, CUNY
Recent developments in the model theory of Abelian lattice-ordered groups

An Abelian lattice-ordered group ($\ell$-group) is an Abelian group with a partial ordering, invariant under translations, that is a lattice ordering. A prototypical example of an $\ell$-group is $C(X)$, the continuous real-valued functions on the topological space $X$ with pointwise operations and ordering. Let $\bf{A}$ be the class of $\ell$-groups, viewed as structures for the first-order language $\mathcal{L}=\{+,-, 0, \wedge, \vee \}$. After giving more background on $\ell$-groups, I will survey what is known about the $\ell$-groups existentially closed (e.c.) in $\bf{A}$, including some new examples I constructed using Fraïssé limits. Then I will discuss some recently published work of Scowcroft concerning the $\ell$-groups e.c. in $\bf{W}^+$, the class of nonzero Archimedean $\ell$-groups with distinguished strong order unit (viewed as structures for $\mathcal{L}_1 = \mathcal{L} \cup \{1 \}$). Building on Scowcroft's results, I will present new axioms for the $\ell$-groups e.c. in $\bf{W}^+$ and show how they allow one to characterize those spaces $X$ for which $(C(X), 1_X)$ is e.c. in $\bf{W}^+$.