Nucleus (order theory)

In mathematics, and especially in order theory, a nucleus is a function ${\displaystyle F}$ on a meet-semilattice ${\displaystyle {\mathfrak {A}}}$ such that (for every ${\displaystyle p}$ in ${\displaystyle {\mathfrak {A}}}$):^[1]

1. ${\displaystyle p\leq F(p)}$
2. ${\displaystyle F(F(p))=F(p)}$
3. ${\displaystyle F(p\wedge q)=F(p)\wedge F(q)}$

Every nucleus is evidently a monotone function.

Frames and locales

Usually, the term nucleus is used in pointless topology (when the semilattice ${\displaystyle {\mathfrak {A}}}$ is a frame).

Proposition: If ${\displaystyle F}$ is a nucleus on a frame ${\displaystyle {\mathfrak {A}}}$, then the poset ${\displaystyle \operatorname {Fix} F}$ of fixed points of ${\displaystyle F}$, with order inherited from ${\displaystyle {\mathfrak {A}}}$, is also a frame.^[2]