I am asked to find some number of totally ordered sets such that no one is isomorphic to a subset of another. To do this I wanted to think about the properties that are preserved for totally ordered sets.
Two examples that I've thought of are the minimal and maximal element. If a totally set has either of these, then any totally ordered set isomorphic to it must also have the same thing.
Another thing is possibly denseness. For example $\mathbb Q$ cannot be isomorphic to a subset of $\mathbb N$ because $\mathbb N$ is not dense in itself. However, I have a gut feeling that something may go wrong for denseness, even though I feel like it is preserved.
Are there any more properties other than this that are preserved under isomorphism between totally ordered sets? Thank you!