Extreme point axioms for closure spaces

A pair (X,@t) of a finite set X and a closure operator @t:2^X->2^X is called a closure space. The class of closure spaces includes matroids as well as antimatroids. Associated with a closure space (X,@t), the extreme point operator ex:2^X->2^X is defined as ex(A)={p|[email protected]?A,[email protected][email protected](A-{p})}. We give characterizations of extreme point operators of closure spaces, matroids and antimatroids, respectively.