There is a considerable literature on maximal sublattices of bounded distributive lattices. In this paper, the authors consider the dual concept of minimal extension. For a distributive bounded lattice \(L\) with Priestley space \((X_{L},\leq _{L},\tau _{L})\), a Priestley space \((X_{E},\leq _{E},\tau _{E})\) is associated with any minimal extension \(E\) of \(L\), which may be obtained from \( (X_{L},\leq _{L},\tau _{L})\) by removing a cover or splitting a point.
For finite \(L\), Birkhoff’s representation solves the problem.
For infinite \(L\), removing a covering pair always results in a compact ordered space. The question is whether or not this space is totally order-disconnected. In topological terms, those covers whose removal results in a Priestley space are characterized.
For \(L\) infinite, if \(a\in X_{L}\), then splitting \(a\) results in a poset \( (X_{E},\leq _{E})\) where \(X_{E}=X_{L}\backslash \{a)\cup \{a_{-},a_{+}\}\). Although it is not always possible to split any element, every Priestley space has special elements that can be. Again in topological terms, special points are identified.
For infinite \(L\), there are at least \(\left| L\right| \) minimal extensions obtained by splitting special points, and examples are given to show that there need be no other minimal extensions.