Interior operator

Closure operators on partially ordered sets

Using the pointwise order on functions between posets, one may alternatively write the extensiveness property as id_P ≤ cl, where id is the identity function. A self-map k that is increasing and idempotent, but satisfies the dual of the extensiveness property, i.e. k ≤ id_P is called a kernel operator, interior operator, or dual closure. As examples, if A is a subset of a set B, then the self-map on the powerset of B given by μ_A(X) = A ∪ X is a closure operator, whereas λ_A(X) = A ∩ X is a kernel operator. The ceiling function from the real numbers to the real numbers, which assigns to every real x the smallest integer not smaller than x, is another example of a closure operator.