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

is an interval order if and only if there exists a bijection from

to a set of real intervals, so

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

-free posets .

[1] Fully written out, this means that for any two pairs of elements

The subclass of interval orders obtained by restricting the intervals to those of unit length, so they all have the form

, is precisely the semiorders.

The complement of the comparability graph of an interval order (

, ≤) is the interval graph

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).

Interval orders' practical applications include modelling evolution of species and archeological histories of pottery styles.

[2][example needed] An important parameter of partial orders is order dimension: the dimension of a partial order

is the least number of linear orders whose intersection is

For interval orders, dimension can be arbitrarily large.

And while the problem of determining the dimension of general partial orders is known to be NP-hard, determining the dimension of an interval order remains a problem of unknown computational complexity.

[3] A related parameter is interval dimension, which is defined analogously, but in terms of interval orders instead of linear orders.

Thus, the interval dimension of a partially ordered set

for which there exist interval orders

The interval dimension of an order is never greater than its order dimension.

[4] In addition to being isomorphic to

-free posets, unlabeled interval orders on

are also in bijection with a subset of fixed-point-free involutions on ordered sets with cardinality

[5] These are the involutions with no so-called left- or right-neighbor nestings where, for any involution

, a left nesting is an

and a right nesting is an

Such involutions, according to semi-length, have ordinary generating function[6] The coefficient of

gives the number of unlabeled interval orders of size

The sequence of these numbers (sequence A022493 in the OEIS) begins

The Hasse diagram 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).
The poset (black Hasse diagram) cannot be part of an interval order: if a is completely right of b , and d overlaps with both a and b , and c is completely right of d , then c must be completely right of b (light gray edge).