A Characterisation of $N$-Free Ordered Groups

Source abstract

We first establish a Gallai-type decomposition for arbitrary two-sided ordered groups. If $G$ is an ordered group and \(g\ne\e\), the least strong module \(S(\e,g)\) containing \(\e\) and $g$ is a convex subgroup, and the subgroups \(S(\e,g)\) form a chain under inclusion. For each such robust subgroup $H$, the union $H^-$ of the proper robust subgroups contained in $H$ is a convex normal subgroup of $H$, and the quotient ordered group $H/H^-$ is either prime, totally ordered, or equality-ordered. Every strong module of $G$ is a coset of a convex subgroup obtained from an initial segment of this chain. Thus the Gallai decomposition acquires a canonical group-theoretic form. We then specialise this decomposition to $N$-free ordered groups. The prime alternative disappears: every quotient $H/H^-$ is either totally ordered or equality-ordered. This yields a canonical reduced two-coloured subgroup chain from which the original order is recovered by a leading-layer rule. Conversely, every reduced conjugation-equivariant subgroup chain satisfying the corresponding least-level and normality conditions, with these two kinds of quotient, defines an $N$-free order by the same rule.