A Characterisation of \(N\)-Free Ordered Groups

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.