On Pseudocomplemented and Stone Ordered Sets

The aim of this paper is to characterize both the pseudocomplemented and Stone ordered sets in a manner similar to that used previously for Boolean and distributive ordered sets. The sublattice G(A) of the Dedekind–Mac Neille completion DM(A) of an ordered set A generated by A is said to be the characteristic lattice of A. We will show that there are distributive pseudocomplemented ordered sets whose characteristic lattices are not pseudocomplemented. We can define a stronger notion of pseudocomplementedness by demanding that both A and G(A) be pseudocomplemented. It turns out that the two concepts are the same for finite and Stone ordered sets.