ABSTRACT
Completely distributive lattices have attracted attention since the early history of lattice theory, notably through the work of Raney. Indeed, the equivalence operates via the spectral theory for completely distributive lattices and the duality theory of continuous posets. The first section is devoted to a careful analysis of the category CD of completely distributive lattices and their morphisms within the framework of Galois adjunctions. The second section actually provides the basic tools of the free construction and discuss their properties. The chapter provides a sufficient measure of generality to derive the existence of free completely distributive lattices as established in the theorems by Dwinger and Markowsky. It concludes the discussion by introducing a functor from continuous posets to continuous lattices with the aid of the free constructions introduced earlier, and the authors show that this functor is the injective hull of a continuous poset.
