$C(\beta\mathbb{N}\setminus\mathbb{N})$ among the Archimedean $\ell$-groups with strong unit
Tom 582 / 2022
Streszczenie
Let $\mathcal{W}^{+}$ be the class of Archimedean lattice-ordered groups with distinguished strong unit $1>0$. This paper studies the $\ell$-groups existentially closed (e.c.) in $\mathcal{W}^{+}$ and those infinitely generic (i.g.) in $\mathcal{W}^{+}$ to cast light upon model-theoretic properties of $C(\beta\mathbb{N}\setminus\mathbb{N})$. After Sections 2–4 obtain infinitary axioms for the $\ell$-groups e.c. in $\mathcal{W}^{+}$, Section 4 shows that when $Z$ and $W$ are Stone spaces of $\aleph_{1}$-saturated atomless Boolean algebras, $C(Z)$ and $C(W)$ are $\aleph_{1}$-existentially saturated in $\mathcal{W}^{+}$ and $C(Z)\equiv_{\infty,\omega_{1}}C(W)$. Section 5 uses this result to show that in every infinitely generic Abelian $\ell$-group, certain interpretable quotient $\ell$-groups are elementarily equivalent to $C(\beta\mathbb{N}\setminus\mathbb{N})$: a result mentioned, but not proved, in the author’s “Algebraically closed and existentially closed Abelian lattice-ordered groups.” Section 6 reaches further conclusions about $C(\beta\mathbb{N}\setminus\mathbb{N})$ by developing the theory of $\ell$-groups e.c., i.g., or finitely generic (f.g.) in $\mathcal{W}^{+}$ and includes a proof that $\ell$-groups i.g. in $\mathcal{W}^{+}$ are elementarily inequivalent to $\ell$-groups f.g. in $\mathcal{W}^{+}$. Section 7 shows that there are continuum-many elementary-equivalence classes of $\ell$-groups e.c. in $\mathcal{W}^{+}$. Section 8 shows that some of the $\ell$-groups from Section 7 may be $\forall\exists$-elementarily embedded into $\ell$-groups i.g. in $\mathcal{W}^{+}$. This result is exploited in Section 9 to show, for example, that every bounded distributive lattice of size at most $\aleph_{1}$ may be nontrivially embedded in the zero-set lattice of $\beta\mathbb{N}\setminus\mathbb{N}$.
