Interval order

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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. More formally, a 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 $x i < x j$ in $P$ exactly when $r_i < \ell_j$ .