The fractional dimension of subsets of Boolean lattices and of cartesian products Original Research Article

The fractional dimension of an ordered set was introduced in Brightwell and Scheinerman (1992). It is an interesting variant of the well studied order dimension and can be considered as a special case of the fractional covering number of hypergraphs. In this paper we provide a geometric interpretation of the fractional dimension and prove three theorems: The fractional dimension of the j and k-level (j < k) of a Boolean lattice is k − j + 2. Second, we deliver a formula for the product of standard orders, and third, we show that the fractional dimension is closed under Dedekind-MacNeille completion. In an appendix the fractional dimensions of 3-irreducible orders are listed.