On the orthographic dimension of definable sets

A formula ϕ(x 1,…,x n) conforms to a partition P of {x 1,…,x n} if it is equivalent to a Boolean combination of formulae that do not have free variables from more than one block of P. We show that if ϕ conforms to two partitions P 1 and P 2 , it also conforms to their greatest lower bound in the partition lattice. As a corollary, we obtain that the concept of orthographic dimension of a constraint-definable set, introduced in the field of constraint databases, is well-defined.