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 . Such posets may be equivalently characterized as those with no induced subposet isomorphic to the pair of two element chains, the free posets .
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.
|Unsolved problem in mathematics:
What is the complexity of determining the order dimension of an interval order?
(more unsolved problems in mathematics)
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-hard, the complexity of determining the order dimension of an interval order is unknown.
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 . These are the involutions with no left or right neighbor nestings where, for an involution on , a left nesting is an such that and a right nesting is an such that .
Such involutions, according to semi-length, have ordinary generating function 
Hence the number of unlabeled interval orders of size is given by the coefficient of in the expansion of .
- Bousquet-Mélou, Mireille; Claesson, Anders; Dukes, Mark; Kitaev, Sergey (2010), "(2+2) free posets, ascent sequences and pattern avoiding permutations", Journal of Combinatorial Theory, Series A, 117 (7): 884–909, doi:10.1016/j.jcta.2009.12.007, MR 2652101.
- Felsner, S. (1992), Interval Orders: Combinatorial Structure and Algorithms (PDF), Ph.D. dissertation, Technical University of Berlin.
- Felsner, S.; Habib, M.; Möhring, R. H. (1994), "On the interplay between interval dimension and dimension" (PDF), SIAM Journal on Discrete Mathematics, 7 (1): 32–40, doi:10.1137/S089548019121885X, MR 1259007.
- Fishburn, Peter C. (1970), "Intransitive indifference with unequal indifference intervals", Journal of Mathematical Psychology, 7 (1): 144–149, doi:10.1016/0022-2496(70)90062-3, MR 0253942.
- Zagier, Don (2001), "Vassiliev invariants and a strange identity related to the Dedekind eta-function", Topology, 40 (5): 945–960, doi:10.1016/s0040-9383(00)00005-7, MR 1860536.
- Fishburn, Peter (1985), Interval Orders and Interval Graphs: A Study of Partially Ordered Sets, John Wiley