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.

## Comments

Joe Zeng

The word "binary predicate" I got from Wikipedia's article on ordered fields, but it looks like it redirects to "binary relation" anyway, so I'll change that.

And "comparison operator" is the terminology in computer science (or at least the one commonly used in programming languages); I wasn't aware that the operator was called an "order" in mathematics in general.