A basic question on the definition of order In the first chapter of Rudin's analysis book "order" on a set is defined as follows :
Let $S$ be a set. An order on $S$ is a relation, denoted by $<$, with the following two properties :
(i) If $x \in S$ and $y \in S$ then one and only one of the statements 
          $$ x < y, x=y, y<x $$ is true.
(ii) If $x,y,z \in S$, then $x < y$ and $y < z$ implies $x<z$. 
How is this different from the usual partial/total order notation. This looks like total order. Why is defining "order" like this ? Moreover, he has not defined $=$ here.

This is slightly different from the standard definition of a total order in that it accounts for $=$ separately, whereas a total order ($\le$, for example), has the following properties:
$\forall x,y \in S$, either  $x \le y$ or $y \le x$. (Totality)
$\forall x,y,z \in S$, if $x \le y$ and $y \le z$, then $x \le z$. (Transitivity)
$\forall x,y \in S$, if $x \le y$ and $y \le x$, then  $x = y$. (Antisymmetry)
, of which this order has only transitivity.
Presumably, Rudin defines "order" like this because it makes some of the proofs in his book more natural than using $\le$ as a total order, or some $\sim$ as a partial order (In which the last two conditions of the total order are still present, but the first is replaced by the weaker condition of reflexivity -- that is, that $\forall x \in S, x \sim x$.)
