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

- 1.
$p\leq F(p)$

- 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 $\mathfrak{A}$ is a frame).

# 1 Some well known results about nuclei

Proposition^{} 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.

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## Mathematics Subject Classification

06B99*no label found*

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