# Nucleus

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

1. 1.

$p\leq F(p)$

2. 2.

$F(F(p))=F(p)$

3. 3.

$F(p\wedge q)=F(p)\wedge F(q)$

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

## 1 Some well known results about nuclei

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.

Title Nucleus Nucleus 2014-12-18 15:34:14 2014-12-18 15:34:14 porton (9363) porton (9363) 1 porton (9363) Definition msc 06B99