The fractional dimension of subsets of Boolean lattices and cartesian products (Q1301736)

The fractional dimension \(\text{fdim} P\) of an ordered set \(P\) was introduced in \textit{G. R. Brightwell} and \textit{E. R. Scheinerman} [Order 9, No. 2, 139-158 (1992; Zbl 0773.06001)]. The main results of the paper under review are the following propositions: 1) if \(C(P)\) is the MacNeille completion of a poset \(P\), then \(\text{fdim} C(P)=\text{fdim} P\) (Theorem 2); 2) if \(B_n(j,k)\) is the Boolean lattice \(B_n\) restricted to the levels \(j,k\) with \(j<k\), then \(\text{fdim} B_n(j,k)= k-j+2\) (Theorem 5); 3) for the standard example \(S_n\), \(n\geq 3\), \(\text{fdim} (S_n \times S_n)= {2n^2\over n+2}\) (Theorem 9). In an appendix the fractional dimensions of 3-irreducible orders are listed.