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, *I*_{1}, being considered less than another, *I*_{2}, if *I*_{1} is completely to the left of *I*_{2}. More formally, a poset is an interval order if and only if there exists a bijection from to a set of real intervals, so, such that for any we have in exactly when .

An interval order defined by unit intervals is a semiorder.

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 containment orders on intervals on the real line (equivalently, the orders of dimension ≤ 2).

Read more about Interval Order: Interval Dimension

### Other articles related to "interval order, interval, order, interval orders, orders":

**Interval Order**- 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 ... 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 ...

### Famous quotes containing the words order and/or interval:

“The world has not to be put in *order*: the world is *order* incarnate. It is for us to put ourselves in unison with this *order*.”

—Henry Miller (1891–1980)

“I was interested to see how a pioneer lived on this side of the country. His life is in some respects more adventurous than that of his brother in the West; for he contends with winter as well as the wilderness, and there is a greater *interval* of time at least between him and the army which is to follow. Here immigration is a tide which may ebb when it has swept away the pines; there it is not a tide, but an inundation, and roads and other improvements come steadily rushing after.”

—Henry David Thoreau (1817–1862)