Polar Orders on Lattices of Real Intervals

Two classes of partial orders are considered and analyzed for the space KC of compact real intervals. The first generalizes the well-known Lower-Upper (LU) partial order, based on the relative position of the intervals’ elements on the real line; the second extends the standard partial order induced by set inclusion. Using mid-point representation, non-empty intervals are associated to points in the ordinary upper half-plane; the two classes of order relations are characterized in terms of two real parameters, appropriately combined to linearly partition the half-plane into four disjoint sliced subsets, which separate comparable from incomparable intervals. We show that pairs of order relations obtained with the same parameters (denote these orders by ⪷ and ⫅, respectively) have an interesting polarity property which allows one to formalize a general setting for analyzing questions of Interval Analysis related to different forms of comparability. In particular, the obtained lattice structures (KC,⪷), (KC,⫅) are detailed and the polarity property is proved for them: if two intervals are incomparable with respect to ⪷ (or ⫅), they are necessarily comparable with respect to ⫅ (or ⪷, respectively). Finally, for any pair of orders (⪷,⫅) with the polarity property, we prove that two total (linear) orders ⪷T and ⫅T are (uniquely) constructed in KC which completely remove incomparable cases. The presentation is supported by several illustrative pictures and some examples.