Bounded countable atomic compactness of ordered groups

We show that whenever $A$ is a monotone $\sigma$-complete dimension group, then $A^+\cup\{\infty\}$ is countably equationally compact, and we show how this property can supply the necessary amount of completeness in several kinds of problems. In particular, if $A$ is a countable dimension group and $E$ is a monotone $\sigma$-complete dimension group, then the ordered group of all relatively bounded homomorphisms from $A$ to $E$ is a monotone $\sigma$-complete dimension group