How to Tell When a Product of Two Partially Ordered Spaces Has a Certain Property

In this paper, we describe how checking whether a given property F is true for a product A1 ×A2 of partially ordered spaces can be reduced to checking several related properties of the original spaces Ai. This result can be useful in the analysis of properties of intervals [a, b] def = {x : a ≤ x ≤ b} over general partially ordered spaces – such as the space of all vectors with component-wise order or the set of all functions with component-wise ordering f ≤ g ⇔ ∀x (f(x) ≤ g(x)). When we consider sets of pairs of such objects A1 × A2, it is natural to define the order on this set in terms of orders in A1 and A2 – this is, e.g., how ordering and intervals are defined on the set IR 2 of all 2-D vectors. This result can also be useful in the analysis of ordered spaces describing different degrees of certainty in expert knowledge.