A totally ordered set $~$(S, \\le)$~$ is a set $~$S$~$ with a comparison operator $~$\\le$~$ that is defined for all members of $~$S$~$\.
Correct me if I'm wrong, but isn't it idiosyncratic to define $~$\leq$~$ as a predicate rather than a relation? I know of at least three books that describe it as a relation: The Joy of Sets by Devlin, Principles of Mathematical Analysis by Rudin, and Introduction to Lattice and Order by Davey and Priestly.
Also, isn't $~$\leq$~$ called an order rather than a comparison?
I bring this up because I would like there to be consistency between this page and the Partially ordered set page. I think both pages should follow the conventions of mathematics.