*-Maximum lattice-ordered groups

Abstract

The category $W$ is archimedean $l$-groups $G$ with distinguished weak order unit $e_G$, with unit-preserving $l$-group homomorphisms. For $G$ in $W$, $G^*$ denotes the convex sub-$l$-group generated by $e_G$. If $G$ satisfies [any isomorphism of $H^*$ with $G^*$ extends to an embedding of $H$ in $G$], then $G$ is called $\ast$-maximum. ($H^*$ and $G^*$ are isomorphic if and only if the unit intervals $[0, e_H]$ and $[0, e_G]$ are isomorphic MV-algebras.) This paper analyzes the property ``$\ast$-maximum'': Several characterizations are given. It is shown that any $\ast$-maximum $G$ has quasi-F Yosida space; this applies to prove a conjecture of the author and J\. Martinez about ``rings of $\omega_1$-quotients.'' It is shown that each $G$ in $W$ has a $\ast$-maximum hull, and this hull is described.