Adjunction of a strong unit to a hyper-archimedean lattice-ordered group

This paper studies conditions in which a hyperarchimedean lattice-ordered group embeds into a hyperarchimedean lattice-ordered group with strong unit. While Conrad and Martinez showed that some hyperarchimedean lattice-ordered groups do not admit such embeddings, Section 3 presents a sufficient condition, in terms of the generalized Boolean algebra of principal \(\ell \)-ideals, for the existence of such an embedding. Section 4 presents new examples of hyperarchimedean lattice-ordered groups not admitting such embeddings, while Section 5 shows that even when such an embedding exists, adjunction of a strong unit may yield non-isomorphic hyperarchimedean extensions. Section 6 shows that if one assumes the existence of weakly compact cardinals, then the sufficient condition from Section 3 is not necessary; and Section 7 studies the logical complexity of the condition “embeddable into a hyperarchimedean lattice-ordered group with strong unit.”