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 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 OrderInterval 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)