# 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 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 $r_i < \ell_j$.

An interval order defined by unit intervals is a semiorder.

The complement of the comparability graph of an interval order ($X$, ≤) is the interval graph $(X, \cap)$.

Interval orders should not be confused with the interval-containment orders, which are the containment orders on intervals on the real line (equivalently, the orders of dimension ≤ 2).

## Interval dimension

The interval dimension of a partial order can be defined as the minimal number of interval order extensions realizing this order, in a similar way to the definition of the order dimension which uses linear extensions. The interval dimension of an order is always less than its order dimension,[1] but interval orders with high dimensions are known to exist. While the problem of determining the order dimension of general partial orders is known to be NP-complete, the complexity of determining the order dimension of an interval order is unknown.[2]

## References

• Fishburn, Peter (1985). Interval Orders and Interval Graphs: A Study of Partially Ordered Sets. John Wiley.
• Gunther Schmidt, 2010. Relational Mathematics. Cambridge University Press, ISBN 978-0-521-76268-7.