Nucleus (order theory)

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

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 $𝔄$ is a frame).

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