An Adjointness Relation for Finite Partition Lattices

We investigate some adjoint functors between categories of order-preserving morphisms. The main adjointness describes the free-like behaviour of partition lattices with respect to distributive lattices in a suitable category. These properties originate from the relationship of the order morphisms with Butler ℬ ⁽¹⁾-groups, a class of torsionfree Abelian groups of finite rank (see [ AV ] for references). The relationship is explained in the introduction.