**Notation:** Let $L_1,L_2,...$ be linearly ordered sets.

- $L_{i}^{-1}$ denotes the reverse linear order of $L_{i}$,
- $L_1+L_2$ denotes the sum of linear orders, i.e. the disjoint union $L_1\cup L_2$ with its natural linear order,
- $\sum_{n\in\omega}L_n$ denotes the sum of linear orders indexed by the natural numbers, i.e. the disjoint union $\bigcup_{n\in\omega}L_n$ with its natural linear order.

**Construction:** Let $S$ be the smallest class of linear orders with the following properties:

- $S$ contains the one-point order $L=\{\ast\}$,
- If $L\in S$, then $L^{-1}\in S$,
- If $L_1,L_2\in S$, then $L_1+L_2\in S$
- If $\{L_n|n\in \omega\}\subseteq S$, then $\sum_{n\in\omega}L_n\in S$.

Restricting to isomorphism classes, $S$ becomes something like a free semigroup of linear orders with involution and infinitary product generated by the one-point order. Using transfinite induction, it is easy to show that every element of $S$ is a countable linear order with countably many Dedekind cuts. However, it is not clear to me that this characterizes the elements of $S$ entirely.

**Question:** Is there a nice characterization of the linear orders in $S$, perhaps in terms of their dedekind cuts?

Grundzüge einer Theorie der geordneten Mengen(Math. Ann. 65) as a reference. $\endgroup$2more comments