On the dimension of the Cartesian product of relations and orders (Q912876)

The triple (G,M,I) is a context if I is a binary relation between the sets G and M. A relation \(F\subset G\times M\) is called a Ferrers relation if \(g_ 1Fm_ 1\) and \(g_ 2Fm_ 2\) implies \(g_ 1Fm_ 2\) or \(g_ 2Fm_ 1\) for all \(g_ 1,g_ 2\in G\) and \(m_ 1,m_ 2\in M\). The Ferrers dimension of a context \(K=(G,M,I)\) (denoted by f dim K) is the smallest number of Ferrers relations \(F_ 1,...,F_ n\) with \(I=\cap F_ i.\) Max\(\{\) f dim K,f dim \(L\}\leq f \dim K\times L\) holds for incidence structures K and L. The author shows that under additional assumptions on K and L, both bounds can be improved. Especially, the square of a three- dimensional ordered set is at least four-dimensional.