Editing Total order (section)

==Orders on the Cartesian product of totally ordered sets==

There are several ways to take two totally ordered sets and extend to an order on the [[Cartesian product]], though the resulting order may only be [[partial order|partial]]. Here are three of these possible orders, listed such that each order is stronger than the next:
* [[Lexicographical order]]: (''a'',''b'') ≤ (''c'',''d'') if and only if ''a'' < ''c'' or (''a'' = ''c'' and ''b'' ≤ ''d''). This is a total order.
* (''a'',''b'') ≤ (''c'',''d'') if and only if ''a'' ≤ ''c'' and ''b'' ≤ ''d'' (the [[product order]]). This is a partial order.
* (''a'',''b'') ≤ (''c'',''d'') if and only if (''a'' < ''c'' and ''b'' < ''d'') or (''a'' = ''c'' and ''b'' = ''d'') (the reflexive closure of the [[Direct product#Direct product of binary relations|direct product]] of the corresponding strict total orders). This is also a partial order.

Each of these orders extends the next in the sense that if we have ''x'' ≤ ''y'' in the product order, this relation also holds in the lexicographic order, and so on. All three can similarly be defined for the Cartesian product of more than two sets.

Applied to the [[vector space]] '''R'''<sup>''n''</sup>, each of these make it an [[ordered vector space]].

See also [[Partially ordered set#Examples|examples of partially ordered sets]].

A real function of ''n'' real variables defined on a subset of '''R'''<sup>''n''</sup> [[Strict weak ordering#Function|defines a strict weak order and a corresponding total preorder]] on that subset.