We say that a group is fully preorderable if every (left- and right-) translation invariant preorder on it can be extended to a translation invariant total preorder. Such groups arise naturally in applications, and relate closely to orderable and fully orderable groups (which were studied extensively since the seminal works of Philip Hall and A. I. Mal’cev in the 1950s). Our first main result provides a purely group-theoretic characterization of fully preorderable groups by means of a condition that goes back to Ohnishi (Osaka Math. J. 2 , 161–164 16 ). In particular, this result implies that every fully orderable group is fully preorderable, but not conversely. Our second main result shows that every locally nilpotent group is fully preorderable, but a solvable group need not be fully preorderable. Several applications of these results concerning the inheritance of full preorderability, connections between full preorderability and full orderability, vector preordered groups, and total extensions of translation invariant binary relations on a group, are provided.

中文翻译：

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完全可预订的团体

如果一个组上的每个（左和右）平移不变预序都可以扩展为平移不变总预序，我们就说该组是完全可预序的。此类群在应用中自然出现，并且与可排序和完全可排序的群密切相关（自 1950 年代 Philip Hall 和 AI Mal'cev 的开创性著作以来，对它们进行了广泛的研究）。我们的第一个主要结果通过可追溯到 Ohnishi (Osaka Math. J. 2 , 161–164 16 ) 的条件提供了完全可预排序群的纯群论表征。特别是，这个结果意味着每个完全可排序的组都是完全可预排序的，但反之则不然。我们的第二个主要结果表明，每个局部幂零群都是完全可预排序的，但可解群不一定是完全可预排序的。