At the start of Baby Rudin (Walter Rudin's Principles of Mathematical Analysis), Rudin defines an order $<$ on a set $S$ as being any relation which obeys the following two axioms:
(1) One and only one of $x<y, x=y, y<x$ is true. (2) If $x<y$ and $y<z$ then $x<z$.
My question is, to what notion of order (in order theory) does this correspond? Is it a total order (which is defined as a relation which is reflexive, transitive, antisymmetric, and strongly connected)?