Why is the Cartesian product $X\times X$ an ordering of $X$?

Question

It should be remarked that the definition of ordering is not very restrictive. For example, $X\times X$ is an ordering of $X,$ but a rather uninteresting one. Relative to this ordering, each member of $X$ is an upper bound, and in fact a supremum, of every subset. The more interesting satisfy the further conditions: if $x$ is less than or equal to $y$ and $y$ is less than or equal to $x,$ than $y=x\,.$ In this case there is at most one supremum for a set, and at most infimum.

This is excerpted from Orderings from John Kelley's General Topology.

Hmm; I stumbled upon the very first part of the excerpt:

What did Kelley mean by that the definition of ordering is not very restrictive?

How is the Cartesian Product $X\times X$ an ordering of $X\,?$

Answer 1

From page 13 of the book:

An ordering (partial ordering, quasi-ordering) is a transitive relation. A relation $<$ orders (partially orders) a set $X$ iff it is transitive on $X$.

Thus, a subset $R$ of $X\times X$ is an ordering of $X$ if and only if for $(x,y)\in R$ and $(y,z)\in R$ we always have $(x,z)\in R$. This is obviously satisfied by $R=X\times X$ and hence $X\times X$ is an ordering of $X$.

Answer 2

What did Kelley mean by that the definition of ordering is not very restrictive?

It means that even things that don't look very much like orderings at all are counted as "orderings". One example is given: $X \times X$.

An ordering is a relation on $X$ -- that is, a subset $R \subseteq X \times X$ -- which is also reflexive and transitive. Then, usually, we write $xRy$ to mean that $(x,y) \in R$. In the case of orders, $R$ is usually written something like $\leq$, so $x \leq y$ is shorthand for $(x,y) \in {\leq}$.

In particular, taking ${\leq} \subseteq X \times X$ to be the entire set $X \times X$, we have that $(x,y) \in {\leq}$ always, or in other words, for any $x,y$ we have $x \leq y$. This does not seem to order much, which is what is meant by "not very restrictive".