Extreme point axioms for closure spaces

Abstract

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