Interval order

Article summary

for a partial order alongside an interval representation of the order.|A partial order on the set {a, b, c, d, e, f} illustrated by its Hasse diagram (left) and a collection of intervals that represents it (right). 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, I 1 , being considered less than another, I 2 , if I 1 is completely to the left of I 2 . More formally, a countable poset P = (X, \leq) is an interval order if and only if there exists a bijection from X to a set of real intervals, so x_i \mapsto (\ell_i, r_i) , such that for any x_i, x_j \in X we have x_i in P exactly when r_i . 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 (2+2) -free posets. Fully written out, this means that for any two pairs of elements a > b and c > d one must have a > d or c > b . The subclass of interval orders obtained by restricting the intervals to those of unit length, so they all have the form (\ell_i, \ell_i + 1) , is precisely the semiorders. The complement of the comparability graph of an interval order ( X , ≤) is the interval graph (X, \cap) .…

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