Nucleus (order theory)

In mathematics, and especially in order theory, a nucleus is a function $$F$$ on a meet-semilattice $$\mathfrak{A}$$ such that (for every $$p$$ in $$\mathfrak{A}$$):


 * 1) $$p \le F(p)$$
 * 2) $$F(F(p)) = F(p)$$
 * 3) $$F(p \wedge q) = F(p) \wedge F(q)$$

Every nucleus is evidently a monotone function.

Frames and locales
Usually, the term nucleus is used in frames and locales theory (when the semilattice $$\mathfrak{A}$$ is a frame).

Proposition: If $$F$$ is a nucleus on a frame $$\mathfrak{A}$$, then the poset $$\operatorname{Fix}(F)$$ of fixed points of $$F$$, with order inherited from $$\mathfrak{A}$$, is also a frame.