Editing Order theory (section)

=== Constructing new orders ===
There are many ways to construct orders out of given orders. The dual order is one example. Another important construction is the [[cartesian product]] of two partially ordered sets, taken together with the [[product order]] on pairs of elements. The ordering is defined by (''a'', ''x'') ≤ (''b'', ''y'') if (and only if) ''a'' ≤ ''b'' and ''x'' ≤ ''y''. (Notice carefully that there are three distinct meanings for the relation symbol ≤ in this definition.) The [[disjoint union]] of two posets is another typical example of order construction, where the order is just the (disjoint) union of the original orders.

Every partial order ≤ gives rise to a so-called [[strict order]] <, by defining ''a'' < ''b'' if ''a'' ≤ ''b'' and not ''b'' ≤ ''a''. This transformation can be inverted by setting ''a'' ≤ ''b'' if ''a'' < ''b'' or ''a'' = ''b''. The two concepts are equivalent although in some circumstances one can be more convenient to work with than the other.