Orders On The Cartesian Product of Partially Ordered Sets
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:
- Lexicographical order: (a,b) ≤ (c,d) if and only if a < c or (a = c and b ≤ d).
- (a,b) ≤ (c,d) if and only if a ≤ c and b ≤ d (the product 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 of the corresponding strict orders).
All three can similarly be defined for the Cartesian product of more than two sets.
Applied to ordered vector spaces over the same field, the result is in each case also an ordered vector space.
See also orders on the Cartesian product of totally ordered sets.
Read more about this topic: Partially Ordered Set
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