Nucleus

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.

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

Some well known results about nuclei

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.