page_title: Partially ordered set
section_title: Orders on the Cartesian product of partially ordered sets
hierarchical_section_title: Partially ordered set / Orders on the Cartesian product of partially ordered sets
context_page_description: In mathematics, especially order theory, a partially ordered set formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation indicating that, for certain pairs of elements in the set, one of the elements precedes the other in the ordering. The relation itself is called a "partial order." The word partial in the names "partial order" and "partially ordered set" is used as an indication that not every pair of elements needs to be comparable. That is, there may be pairs of elements for which neither element precedes the other in the poset. Partial orders thus generalize total orders, in which every pair is comparable.
  Formally, a partial order is any binary relation that is reflexive, antisymmetric, and transitive.
  One familiar example of a partially ordered set is a collection of people ordered by genealogical descendancy. Some pairs of people bear the descendant-ancestor relationship, but other pairs of people are incomparable, with neither being a descendant of the other.
  A poset can be visualized through its Hasse diagram, which depicts the ordering relation.
context_section_description: 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 figures):
  the lexicographical order:   (a,b) ≤ (c,d) if a < c or (a = c and b ≤ d);
  the product order:    (a,b) ≤ (c,d) if a ≤ c and b ≤ d;
  the reflexive closure of the direct product of the corresponding strict orders:    (a,b) ≤ (c,d) if (a < c and b < d) or (a = c and b = d).
  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.
caption: Reflexive closure of strict direct product order on ℕ×ℕ. Elements covered by (3,3) and covering (3,3) are highlighted in green and red, respectively.