A closure operator on a partially ordered set (P, ≤) is a function c: P → P that is:
- Monotone: if a ≤ b then c(a) ≤ c(b)
- Extensive: a ≤ c(a) for all a
- Idempotent: c(c(a)) = c(a) for all a
The fixed points of a closure operator (elements a where c(a) = a) form a complete lattice. Closure operators appear throughout mathematics: topological closure, algebraic closure, deductive closure, and many others share this structure.
For a curriculum introduction, see the existing learn-closure-operators skill. In the relationality derivation, the concept of closure appears philosophically at step 3 (self-sustaining closure) and formally at step 16 as Nucleus — a closure operator on the Heyting algebra. See Closure in the Derivation for how these connect.