Order dimension

The dimension of a poset P is the least integer t for which there exists a family

${\displaystyle {\mathcal {R}}=(<_{1},\dots ,<_{t})}$ 

of linear extensions of P so that, for every x and y in P, x precedes y in P if and only if it precedes y in all of the linear extensions, if any such t exists. That is,

${\displaystyle P=\bigcap {\mathcal {R}}=\bigcap _{i=1}^{t}<_{i}.}$ 

An alternative definition of order dimension is the minimal number of total orders such that P embeds into their product with componentwise ordering i.e. ${\displaystyle x\leq y}$  if and only if ${\displaystyle x_{i}\leq y_{i}}$  for all i (Hiraguti 1955, Milner & Pouzet 1990).

A family ${\displaystyle {\mathcal {R}}=(<_{1},\dots ,<_{t})}$  of linear orders on X is called a realizer of a poset P = (X, <_P) if

${\displaystyle <_{P}=\bigcap {\mathcal {R}}}$ ,

which is to say that for any x and y in X, x <_P y precisely when x <₁ y, x <₂ y, ..., and x <_t y. Thus, an equivalent definition of the dimension of a poset P is "the least cardinality of a realizer of P."

It can be shown that any nonempty family R of linear extensions is a realizer of a finite partially ordered set P if and only if, for every critical pair (x,y) of P, y <_i x for some order <_i in R.

Let n be a positive integer, and let P be the partial order on the elements a_i and b_i (for 1 ≤ i ≤ n) in which a_i ≤ b_j whenever i ≠ j, but no other pairs are comparable. In particular, a_i and b_i are incomparable in P; P can be viewed as an oriented form of a crown graph. The illustration shows an ordering of this type for n = 4.

Then, for each i, any realizer must contain a linear order that begins with all the a_j except a_i (in some order), then includes b_i, then a_i, and ends with all the remaining b_j. This is so because if there were a realizer that didn't include such an order, then the intersection of that realizer's orders would have a_i preceding b_i, which would contradict the incomparability of a_i and b_i in P. And conversely, any family of linear orders that includes one order of this type for each i has P as its intersection. Thus, P has dimension exactly n. In fact, P is known as the standard example of a poset of dimension n, and is usually denoted by S_n.

The partial orders with order dimension two may be characterized as the partial orders whose comparability graph is the complement of the comparability graph of a different partial order (Baker, Fishburn & Roberts 1971). That is, P is a partial order with order dimension two if and only if there exists a partial order Q on the same set of elements, such that every pair x, y of distinct elements is comparable in exactly one of these two partial orders. If P is realized by two linear extensions, then partial order Q complementary to P may be realized by reversing one of the two linear extensions. Therefore, the comparability graphs of the partial orders of dimension two are exactly the permutation graphs, graphs that are both themselves comparability graphs and complementary to comparability graphs.

The partial orders of order dimension two include the series-parallel partial orders (Valdes, Tarjan & Lawler 1982). They are exactly the partial orders whose Hasse diagrams have dominance drawings, which can be obtained by using the positions in the two permutations of a realizer as Cartesian coordinates.

It is possible to determine in polynomial time whether a given finite partially ordered set has order dimension at most two, for instance, by testing whether the comparability graph of the partial order is a permutation graph. However, for any k ≥ 3, it is NP-complete to test whether the order dimension is at most k (Yannakakis 1982).

The incidence poset of any undirected graph G has the vertices and edges of G as its elements; in this poset, x ≤ y if either x = y or x is a vertex, y is an edge, and x is an endpoint of y. Certain kinds of graphs may be characterized by the order dimensions of their incidence posets: a graph is a path graph if and only if the order dimension of its incidence poset is at most two, and according to Schnyder's theorem it is a planar graph if and only if the order dimension of its incidence poset is at most three (Schnyder 1989).

For a complete graph on n vertices, the order dimension of the incidence poset is ${\displaystyle \Theta (\log \log n)}$  (Hoşten & Morris 1999). It follows that all simple n-vertex graphs have incidence posets with order dimension ${\displaystyle O(\log \log n)}$ .

A generalization of dimension is the notion of k-dimension (written ${\displaystyle {\textrm {dim}}_{k}}$ ) which is the minimal number of chains of length at most k in whose product the partial order can be embedded. In particular, the 2-dimension of an order can be seen as the size of the smallest set such that the order embeds in the inclusion order on this set.

• Interval dimension

• Baker, K. A.; Fishburn, P.; Roberts, F. S. (1971), "Partial orders of dimension 2", Networks, 2 (1): 11–28, doi:10.1002/net.3230020103.
• Dushnik, Ben; Miller, E. W. (1941), "Partially ordered sets", American Journal of Mathematics, 63 (3): 600–610, doi:10.2307/2371374, JSTOR 2371374.
• Hiraguti, Tosio (1955), "On the dimension of orders" (PDF), The Science Reports of the Kanazawa University, 4 (1): 1–20, MR 0077500.
• Hoşten, Serkan; Morris, Walter D. Jr. (1999), "The order dimension of the complete graph", Discrete Mathematics, 201 (1–3): 133–139, doi:10.1016/S0012-365X(98)00315-X, MR 1687882.
• Milner, E. C.; Pouzet, M. (1990), "A note on the dimension of a poset", Order, 7 (1): 101–102, doi:10.1007/BF00383178, MR 1086132, S2CID 123485792.
• Schnyder, W. (1989), "Planar graphs and poset dimension", Order, 5 (4): 323–343, doi:10.1007/BF00353652, S2CID 122785359.
• Trotter, William T. (1992), Combinatorics and partially ordered sets: Dimension theory, Johns Hopkins Series in the Mathematical Sciences, The Johns Hopkins University Press, ISBN 978-0-8018-4425-6.
• Valdes, Jacobo; Tarjan, Robert E.; Lawler, Eugene L. (1982), "The recognition of series parallel digraphs", SIAM Journal on Computing, 11 (2): 298–313, doi:10.1137/0211023.
• Yannakakis, Mihalis (1982), "The complexity of the partial order dimension problem", SIAM Journal on Algebraic and Discrete Methods, 3 (3): 351–358, doi:10.1137/0603036.