Every left-invariant ordering of a group is either discrete, meaning there is a least element greater than the identity, or dense. Corresponding to this dichotomy, the spaces of left, Conradian, and bi-orderings of a group are naturally partitioned into two subsets. This note investigates the structure of this partition, specifically the set of dense orderings of a group and its closure within the space of orderings. We show that for bi-orderable groups this closure will always contain the space of Conradian orderings---and often much more. In particular, the closure of the set of dense orderings of the free group is the entire space of left-orderings.

中文翻译：

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一组左序空间中的稠序

一个群的每个左不变排序要么是离散的，这意味着有一个最小元素大于身份，要么是稠密的。对应于这种二分法，一个组的左序、康弧式和双序空间自然地划分为两个子集。这篇笔记研究了这个分区的结构，特别是一个群的密集排序集及其在排序空间内的闭包。我们表明，对于双序群，这个闭包将始终包含康弧度排序的空间——而且通常更多。特别地，自由群的稠密序集的闭包是整个左序空间。