Work of Linnell shows that the space of left-orderings of a group is either finite or uncountable, and in the case that the space is finite, the isomorphism type of the group is known—it is what is known as a Tararin group. By defining semiconjugacy of circular orderings in a general setting (that is, for arbitrary circular orderings of groups that may not act on $S^1$), we can view the subspace of left-orderings of any group as a single semiconjugacy class of circular orderings. Taking this perspective, we generalize the result of Linnell, to show that every semiconjugacy class of circular orderings is either finite or uncountable, and when a semiconjugacy class is finite, the group has a prescribed structure. We also investigate the space of left-orderings as a subspace of the space of circular orderings, addressing a question of Baik and Samperton.

Linnell 的工作表明，一个群的左序空间要么是有限的，要么是不可数的，在空间有限的情况下，该群的同构类型是已知的——它就是所谓的 Tararin 群。通过在一般设置中定义循环排序的半共轭（即，对于可能不作用于的组的任意循环排序）${秒}^1$)，我们可以将任何组的左序子空间视为一个循环排序的单个半共轭类。从这个角度出发，我们概括了 Linnell 的结果，以表明循环排序的每个半共轭类要么是有限的，要么是不可数的，当一个半共轭类是有限的时，该群具有规定的结构。我们还研究了作为循环排序空间的子空间的左排序空间，解决了 Baik 和 Samperton 的问题。