Dimension Theory for Ordered Sets

Abstract

In 1930, E. Szpilrajn proved that any order relation on a set X can be extended to a linear order on X. It also follows that any order relation is the intersection of its linear extensions. B. Dushnik and E.W. Miller later defined the dimension of an ordered set P = 〈X;≤〉 to be the minimum number of linear extensions whose intersection is the ordering ≤.

For a cardinal m, ${\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{2} ^m}$ denotes the subsets of m, ordered by inclusion. As the notation indicates, ${\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{2} ^m}$ is a product of 2-element chains (linearly ordered sets). Any poset 〈X;≤〉 with ∣x∣ ≤ m can be embedded in ${\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{2} ^m}$. O. Ore proved that the dimension of a poset P is the least number of chains whose product contains P as a subposet. He also showed that the product of m nontrivial chains has dimension m. In particular, ${\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{2} ^m}$ has dimension m, a result of H. Komm. Thus, every cardinal is the dimension of some poset.

It is usually very difficult to calculate the dimension of any “standard” poset. However, dimension can be related to other parameters of a poset. For example, the dimension of a finite poset does not exceed the size of any maximal antichain. Also, T. Hiraguchi showed that any poset of dimension d ≥ 3 has at least 2d elements. Moreover, any integer ≥ 2d is the size of some poset of dimension d.

Let d be a positive integer. A poset is d-irreducible if it has dimension d and removal of any element lowers its dimension. Any poset whose dimension is at least d contains a d-irreducible subposet. Although there is only one 2-irreducible poset, there are infinitely many d-irreducible posets whenever d ≥ 3. The set of all 3-irreducible posets was independently determined by D. Kelly and W.T. Trotter, Jr. and J.I. Moore, Jr. There is a 3-irreducible poset of any size n not excluded by Hiraguchi; i.e., for any n ≥ 6. However, R.J. Kimble, Jr. has shown that a d-irreducible poset cannot have size 2d + 1 when d ≥ 4. If d ≥ 4 and n ≥ 2d but n ≠ 2d + 1, then there is a d-irreducible poset of size n.

A finite poset is planar if its diagram can be drawn in the plane without any crossing of lines. Planar posets have arbitrary finite dimension. However, K.A. Baker showed that a finite lattice is planar exactly when its dimension does not exceed 2. He also showed that the completion of a poset is a lattice that has the same dimension as the poset. Baker’s results and three papers of D. Kelly and I. Rival were used to obtain the list of 3-irreducible posets.

The approach of W.T. Trotter and J.I. Moore, Jr. rested on the observation of Dushnik and Miller that a poset has dimension at most 2 if and only if its incomparability graph is a comparability graph. T. Gallai’s characterization of comparability graphs in terms of excluded subgraphs was then applied.

Several other connections between dimension theory for posets and graph theory have been established. For example, posets with the same comparability graph have the same dimension. C.R. Platt reduced the planarity of a finite lattice to the planarity of an undirected graph obtained by adding an edge to its diagram.