A basic question on the definition of order

Question

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.

Answer 1

For the =, you can define it by negating the other two cases. That is, = means not $<$ and not $>$

Answer 2

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 $\forall x \in S, x \sim x$.)

Answer 3

The root of your difficulty seems to be that people use different conventions for defining "order". The first issue arises in the defintion of "partial order", by which some people mean a reflexive, transitive, and antisymmetric relation $\leq$, while other people mean an irreflexive, transitive relation $<$. Given a partial order on a set $A$ in either sense, one can easily define the corresponding order in the other sense, by adjoining or removing the pairs $(a,a)$ for all $a\in A$. So people don't usually worry too much about the distinction, but technically they are two different notions of order. I'm accustomed to calling the reflexive version a "partial order" and the irreflexive version a "strict partial order", but there are people who prefer to use the shorter name "partial order" for the irreflexive version.

Total (or linear) orders are then defined in the reflexive version by requiring $a\leq b$ or $b\leq a$ for all $a,b\in A$, and in the irreflexive version by instead requiring $a<b$ or $a=b$ or $b<a$.
Again, one can convert either sort of total order to the other sort, just as with partial orders.

To further increase the confusion, some people use "order" (without any adjective) to mean a partial order, while others use it to mean a total order. So the final result is that "order" can have any of four meanings. You just have to get used to that; you'll get no suport for denouncing Rudin just because he chose a convention different from the one you learned first.