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 $\mathrm{Fix}(F)$ of fixed points of $F$, with order inherited from $\mathfrak{𝔄}$, is also a frame.