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 countable poset ${\displaystyle P=(X,\leq )}$ is an interval order if and only if there exists a bijection from ${\displaystyle X}$ to a set of real intervals, so ${\displaystyle x_{i}\mapsto (\ell _{i},r_{i})}$, such that for any ${\displaystyle x_{i},x_{j}\in X}$ we have ${\displaystyle x_{i}<x_{j}}$ in ${\displaystyle P}$ exactly when ${\displaystyle r_{i}<\ell _{j}}$.

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 ${\displaystyle (2+2)}$-free posets .^[1] Fully written out, this means that for any two pairs of elements ${\displaystyle a>b}$ and ${\displaystyle c>d}$ one must have ${\displaystyle a>d}$ or ${\displaystyle c>b}$.

The subclass of interval orders obtained by restricting the intervals to those of unit length, so they all have the form ${\displaystyle (\ell _{i},\ell _{i}+1)}$, is precisely the semiorders.

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

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