How small can a lattice of order-dimension n be? (Q1820180)

This paper is concerned with the dimension of two particular classes of lattices, namely partition lattices and linear lattices. The main motivation was provided by the following question: For a lattice L, does dim L\(=n\) always imply \(| L| \geq 2^ n?\) The answer is in the negative. Indeed, suitable lower bounds are established on the dimension of both partition and linear lattices that show that from some n on both classes of lattices fail to obey the bound \(2^ n\). - In the case of partition lattices, the dimension is determined up to an absolute constant. For \(L_ n\), the linear lattice over GF(2), the authors succeed in determining the dimension up to a factor C/n for an absolute constant C.