Editing Partially ordered set (section)

=== Orders on the Cartesian product of partially ordered sets ===
{{multiple image
| align = right
| dirction = horizontal
| total_width = 550
| image1 = Lexicographic order on pairs of natural numbers.svg
| caption1 = '''Fig. 4a''' Lexicographic order on <math>\N \times \N</math>
| image2 = N-Quadrat, gedreht.svg|
| caption2 = '''Fig. 4b''' Product order on <math>\N \times \N</math>
| image3 = Strict product order on pairs of natural numbers.svg|
| caption3 = '''Fig. 4c''' Reflexive closure of strict direct product order on <math>\N \times \N.</math> Elements [[#Formal definition|covered]] by {{nowrap|(3, 3)}} and covering {{nowrap|(3, 3)}} are highlighted in green and red, respectively.
}}
In order of increasing strength, i.e., decreasing sets of pairs, three of the possible partial orders on the [[Cartesian product]] of two partially ordered sets are (see Fig.&nbsp;4):
* the [[lexicographical order]]: &nbsp; {{nowrap|(''a'', ''b'') ≤ (''c'', ''d'')}} if {{nowrap|''a'' < ''c''}} or ({{nowrap|1=''a'' = ''c''}} and {{nowrap|''b'' ≤ ''d''}});
* the [[product order]]:  &nbsp; (''a'', ''b'') ≤ (''c'', ''d'') if ''a'' ≤ ''c'' and ''b'' ≤ ''d'';
* the [[reflexive closure]] of the [[Direct product#Direct product of binary relations|direct product]] of the corresponding strict orders:  &nbsp; {{nowrap|(''a'', ''b'') ≤ (''c'', ''d'')}} if ({{nowrap|''a'' < ''c''}} and {{nowrap|''b'' < ''d''}}) or ({{nowrap|1=''a'' = ''c''}} and {{nowrap|1=''b'' = ''d''}}).

All three can similarly be defined for the Cartesian product of more than two sets.

Applied to [[ordered vector space]]s over the same [[Field (mathematics)|field]], the result is in each case also an ordered vector space.

See also [[Total order#Orders on the Cartesian product of totally ordered sets|orders on the Cartesian product of totally ordered sets]].