Orders On The Cartesian Product of Totally Ordered Sets
In order of increasing strength, i.e., decreasing sets of pairs, three of the possible orders on the Cartesian product of two totally ordered sets are:
- 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 of the corresponding strict total orders). This is also a partial order.
All three can similarly be defined for the Cartesian product of more than two sets.
Applied to the vector space Rn, each of these make it an ordered vector space.
See also examples of partially ordered sets.
A real function of n real variables defined on a subset of Rn defines a strict weak order and a corresponding total preorder on that subset.
Read more about this topic: Total Order
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