Prevalent and shy sets

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, 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.

References

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• Gunther Schmidt, 2010. Relational Mathematics. Cambridge University Press, ISBN 978-0-521-76268-7.