On Cartesian products of compact lines

This is joint work with Gonzalo Martinez Cervantes. By a compact line we mean a compact space which topology is defined by a linear order. Answering a problem posed by Mardesic, we show that n-fold product of compact lines cannot be continuously mapped onto a product of n+1 nonmetrizable spaces. Our proof makes use of a certain topological invariant that we call free dimension. That free dimension may be applied to some questions concerning embeddability of free products of Boolean algebras.