The completion of a poset in a lattice of antichains

It is well known that given a poset, X, the lattice of order ideals of X, 〈I(X),⊆〉, is a completion of X via the order-embedding φ : X ↪→ I(X) where φ(x) = ↓x. Herein we define a lattice of antichains in X, 〈A(X),4〉, and prove it is isomorphic to 〈I(X),⊆〉. We establish the “join” and “meet” operations of the lattice, and present results for 〈A(X),4〉 analogous to standard results for 〈I(X),⊆〉, including Birkhoff’s Representation Theorem for finite distributive lattices and a Dedekind-MacNeille-style completion using antichains. We also discuss the relevance and application of completions using antichains to access control in computer science, in particular with reference to role-based access control and to modelling conflict of interest policies.