Distributivity of the Normal Completion, and Its Priestley Representation

The normal, or Dedekind-MacNeille, completion δ(L) of a distributive lattice L need not be distributive. However, δ(L) does contain a largest distributive sublattice β(L) containing L, and δ(L) is distributive if and only if β(L) is complete if and only if δ(L) = β(L). In light of these facts, it may come as a surprise to learn that β(L) was developed (in [1]) for reasons having nothing to do with distributivity. In fact, the cuts of β(L) can be readily identified as those having the property we here term exactness. This provides a useful criterion for testing whether the normal completion of a given lattice is distributive. We illustrate the utility of this criterion by providing a simple demonstration that the normal completion of a Heyting algebra is distributive. We prove these facts by simple arguments from first princples, and then bring out the geometry of the situation by developing the construct in Priestley spaces. While the elements of L appear as clopen up-sets of the (ordered) space, the elements of both extensions δ(L) and β(L) are manifested as well defined more general types of open up-sets.