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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 P = {\displaystyle P=} is an interval order if and only ifthere exists a bijection from X {\displaystyle X} to a set of real intervals,so x i ↦ {\displaystyle x_{i}\mapsto } ,such that for any x i , x j ∈ X {\displaystyle x_{i},x_{j}\in X} we have x i < x j {\displaystyle x_{i} The subclass of interval orders obtained by restricting the intervals to those of unit length, so they all have the form {\displaystyle } , is precisely the semiorders. The complement of the comparability graph of an interval order is the interval graph {\displaystyle }. Interval orders should not be confused with the interval-containment orders, which are the inclusion orders on intervals on the real line.