Interval order
In mathematics, especially order theory,the interval order for a collection of intervals on the real lineis 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 ifthere exists a bijection from to a set of real intervals,so ,such that for any we have in exactly when .Such posets may be equivalentlycharacterized as those with no induced subposet isomorphic to thepair of two-element chains, in other words as the -free posets. 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). (Wikipedia).
