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 Æ and Ň, 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 pKC,Æq, pKC, Ňq are detailed and the polarity property is proved for them: if two intervals are incomparable with respect to Æ (or Ň), they are necessarily comparable with respect to Ň (or Æ, respectively). Finally, for any pair of orders pÆ, Ňq with the polarity property, we prove that two total (linear) orders ÆT and ŇT are (uniquely) constructed in KC which completely remove incomparable cases. The presentation is supported by several illustrative pictures and some examples.