Nucleus
In order theory, a nucleus is a function $F$ on a meetsemilattice $\U0001d504$ such that (for every $p$ in $\U0001d504$):

1.
$p\le F(p)$

2.
$F(F(p))=F(p)$

3.
$F(p\wedge q)=F(p)\wedge F(q)$
Usually, the term nucleus is used in frames and locales theory (when the semilattice $\U0001d504$ is a frame).
1 Some well known results about nuclei
Proposition^{} If $F$ is a nucleus on a frame $\U0001d504$, then the poset $\mathrm{Fix}(F)$ of fixed points^{} of $F$, with order inherited from $\U0001d504$, is also a frame.
Title  Nucleus 

Canonical name  Nucleus 
Date of creation  20141218 15:34:14 
Last modified on  20141218 15:34:14 
Owner  porton (9363) 
Last modified by  porton (9363) 
Numerical id  1 
Author  porton (9363) 
Entry type  Definition 
Classification  msc 06B99 