Interval order

In mathematics, especially order theory, the interval order for a collection of intervals on the real line is the partial order corresponding to their left-to-right precedence relation—one interval, I1, being considered less than another, I2, if I1 is completely to the left of I2. More formally, a countable poset is an interval order if and only if there exists a bijection from to a set of real intervals, so , such that for any we have in exactly when . Such posets may be equivalently characterized as those with no induced subposet isomorphic to the pair of two-element chains, in other words as the -free posets Fully written out, this means that for any two pairs of elements and one must have or . The subclass of interval orders obtained by restricting the intervals to those of unit length, so they all have the form , is precisely the semiorders. The complement of the comparability graph of an interval order (, ≤) is the interval graph . Interval orders should not be confused with the interval-containment orders, which are the inclusion orders on intervals on the real line (equivalently, the orders of dimension ≤ 2). An important parameter of partial orders is order dimension: the dimension of a partial order is the least number of linear orders whose intersection is . For interval orders, dimension can be arbitrarily large. And while the problem of determining the dimension of general partial orders is known to be NP-hard, determining the dimension of an interval order remains a problem of unknown computational complexity. A related parameter is interval dimension, which is defined analogously, but in terms of interval orders instead of linear orders. Thus, the interval dimension of a partially ordered set is the least integer for which there exist interval orders on with exactly when and . The interval dimension of an order is never greater than its order dimension.