Closure operators and extreme-point operators

An operator 2^E -> 2^E is called a closure operator when it is extensive, monotone, and idempotent. A closure operator satisfying the exchange axiom is a matroid closure operator. A closure operator satisfying the anti-exchange axiom is the closure operator of a convex geometry. A matroid and a convex geometry are also defined in terms of extreme-point operators. The extreme-point operator gives the extreme points of the convex-hull when we consider an affine point configuration. It is known that there is a one-to-one correspondence between closure operators and extreme-point operators. I will introduce related results in my talk. Finally, we characterize some classes in terms of forbidden lattice intervals.