6 - Order codings and dimensions

Summary

We have several times considered codings (Definition 3.1) from an ordered set to another one. Indeed when an ordered set has a complex structure, it is natural to try to represent it in a simpler ordinal structure. In this chapter, we choose direct products of chains as the simple ordered structures (see Section 1.5.2). A coding of the ordered set P will then be a map sending P to an ordered subset, isomorphic to P, of such a product. This notion is particularized when adding conditions on the sizes of the chains. Thus, when all chains have size k (with k a fixed integer), as always assumed in the sequel, we will speak of a k-coding. The minimum number of chains required for the existence of a k-coding of P will be called the k-dimension of P.

In the first section, we study the 2-codings of an ordered set P, and the associated 2-dimension – also called Boolean dimension. Indeed, a 2-coding of P, i.e., a coding from P to a direct product of p chains of size 2, is equivalent to a coding from P to the Boolean lattice of subsets of a set of size p (such a coding may also be called Boolean). The Boolean dimension of an ordered set P is thus the minimum size of a set a family of subsets of which, ordered by set inclusion, reproduces exactly (i.e., is isomorphic to) P.