发表机构
Royal Military College of Canada（加拿大皇家军事学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过Gallai型分解刻画N-自由有序群，证明其商仅有全序或等序两种类型，并给出由约化双色子群链恢复序的充要条件。
AI 中文摘要
我们首先对任意双侧有序群建立一种Gallai型分解。若G是有序群且g≠e，则包含e和g的最小强模S(e,g)是一个凸子群，且这些子群S(e,g)在包含关系下构成一个链。对于每个这样的稳健子群H，包含在H中的真稳健子群的并集H⁻是H的一个凸正规子群，且商有序群H/H⁻要么是素序的、要么是全序的、要么是等序的。G的每个强模都是从这个链的一个初始段得到的凸子群的陪集。因此，Gallai分解获得了一个典范的群论形式。然后我们将此分解专门化到N-自由有序群。素序的可能性消失：每个商H/H⁻要么是全序的、要么是等序的。这产生了一个典范的约化双色子群链，原始序通过一个前导层规则从中恢复。反之，每个满足相应最底层和正规性条件、且具有这两种商的约化共轭等变子群链，都通过相同规则定义一个N-自由序。
英文摘要
We study Gallai decomposition for groups equipped with two-sided invariant partial orders. The key algebraic step extends to arbitrary binary relations compatible with the group operation: if all left and right translations preserve a binary relation $ρ$, then every least strong module $S_ρ(\e,g)$, $g\ne\e$, is a subgroup. For a partial order this subgroup is convex. Thus the robust modules through the identity of an ordered group form a canonical chain of convex subgroups, with each canonical factor $H/H^-$ prime, totally ordered, or equality-ordered. We characterize exactly the subgroups that are modules, show that they form a complete sublattice of the subgroup lattice, establish overlap and inheritance results for arbitrary subgroups, and prove compatibility with quotients by normal strong subgroups. For $N$-free ordered groups the prime factors disappear. Using the robust-module decomposition of cographs, we characterize all two-sided invariant $N$-free partial orders by reduced admissible two-coloured subgroup chains, with totally ordered and equality-ordered canonical factors; the order is determined by the first nontrivial factor of each element. We also determine how the canonical decomposition restricts to arbitrary subgroups, characterize finite width and prove width divisibility for subgroups, and show that every reduced two-coloured chain is realized by an $N$-free ordered abelian group whose canonical factors are isomorphic to $\mathbb Z$.
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