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 = to be the minimum number of linear extensions whose intersection is the ordering S. , m For a cardinal m, 2 denotes the subsets of m, ordered by inclusion. As the notation indicates, Zm is a product of 2-element chains (linearly ordered sets). Any poset with Ixi s m can be embedded in 2m. O. Ore proved that the dimension of a poset P is the least nUllber of chains whose product contains P as a subposet. He also showed that the prod'~ct of m nontrivial chains has dimension m. In particular, 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.