Extreme point axioms for closure spaces

A pair (X,τ)(X,τ) of a finite set X   and a closure operator τ:2X→2Xτ:2X→2X is called a closure space. The class of closure spaces includes matroids as well as antimatroids. Associated with a closure space (X,τ)(X,τ), the extreme point operator ex:2X→2Xex:2X→2X is defined as ex(A)={p|p∈A,p∉τ(A-{p})}ex(A)={p|p∈A,p∉τ(A-{p})}. We give characterizations of extreme point operators of closure spaces, matroids and antimatroids, respectively.