The poset of closure systems on an infinite poset: detachability and semimodularity

Abstract

Closure operators on a poset can be characterized by the corresponding closure systems. It is known that in a directed complete partial order (DCPO), in particular in any finite poset, the collection of all closure systems is closed under arbitrary intersection and has a “detachability” or “anti-matroid” property, which implies that the collection of all closure systems is a lower semimodular complete lattice (and dually, the closure operators form an upper semimodular complete lattice).

After reviewing the history of the problem, we generalize these results to the case of an infinite poset where closure systems do not necessarily constitute a complete lattice; thus the notions of lower semimodularity and detachability are extended accordingly. We also give several examples showing that many properties of closure systems on a complete lattice do not extend to infinite posets.