2 results about "Product order" patented technology

In mathematics, given two partially ordered sets A and B, the product order (also called the coordinatewise order or componentwise order) is a partial ordering on the cartesian product A × B. Given two pairs (a₁, b₁) and (a₂, b₂) in A × B, one defines (a₁, b₁) ≤ (a₂, b₂) if and only if a₁ ≤ a₂ and b₁ ≤ b₂. Another possible ordering on A × B is the lexicographical order, which is a total ordering. However the product order of two totally ordered sets is not in general total; for example, the pairs (0, 1) and (1, 0) are incomparable in the product order of the ordering 0 < 1 with itself.