Interval order#Interval orders and dimension

Image:Interval order, Hasse diagram and interval realization.png 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).]]

File:Critical pair (order theory).svg

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 $P = (X, \leq)$ is an interval order if and only if

there exists a bijection from $X$ to a set of real intervals,

so $x_i \mapsto (\ell_i, r_i)$ ,

such that for any $x_i, x_j \in X$ we have

$x_i < x_j$ in $P$ exactly when $r_i < \ell_j$ .

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 $(2+2)$ -free posets

.{{harvtxt|Fishburn|1970}} Fully written out, this means that for any two pairs of elements $a > b$ and $c > d$ one must have $a > d$ or $c > b$ .

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

The complement of the comparability graph of an interval order ( $X$ , ≤)

is the interval graph $(X, \cap)$ .

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).

Interval orders' practical applications include modelling evolution of species and archeological histories of pottery styles.{{sfn|Davey|Priestley|2002|page=5}}{{Example needed|date=December 2024}}

Interval orders and dimension

{{unsolved|mathematics|What is the complexity of determining the order dimension of an interval order?}}

An important parameter of partial orders is order dimension: the dimension of a partial order $P$ is the least number of linear orders whose intersection is $P$ . 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.{{harvtxt|Felsner|1992|page=47}}

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 $P = (X, \leq)$ is the least integer $k$ for which there exist interval orders $\preceq_1, \ldots, \preceq_k$ on $X$ with $x \leq y$ exactly when $x \preceq_1 y, \ldots,$ and $x \preceq_k y$ .

The interval dimension of an order is never greater than its order dimension.{{sfnp|Felsner|Habib|Möhring|1994}}

Combinatorics

In addition to being isomorphic to $(2+2)$ -free posets,

unlabeled interval orders on $[n]$ are also in bijection

with a subset of fixed-point-free involutions

on ordered sets with cardinality $2n$

.{{harvtxt|Bousquet-Mélou|Claesson|Dukes|Kitaev|2010}} These are the

involutions with no so-called left- or right-neighbor nestings where, for any involution

$f$ on $[2n]$ , a left nesting is

an $i \in [2n]$ such that $i < i+1 < f(i+1) < f(i)$ and a right nesting is an $i \in [2n]$ such that

$f(i) < f(i+1) < i < i+1$ .

Such involutions, according to semi-length, have ordinary generating function{{harvtxt|Zagier|2001}}

: $F(t) = \sum_{n \geq 0} \prod_{i = 1}^n (1-(1-t)^i).$

The coefficient of $t^n$ in the expansion of $F(t)$ gives the number of unlabeled interval orders of size $n$ . The sequence of these numbers {{OEIS|id=A022493}} begins

:1, 2, 5, 15, 53, 217, 1014, 5335, 31240, 201608, 1422074, 10886503, 89903100, 796713190, 7541889195, 75955177642, …

Notes

References

{{citation

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{{citation

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{{citation

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{{cite book

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|edition=2nd}}

Interval order#Interval orders and dimension

Interval orders and dimension

Combinatorics

Notes

References

Further reading