Nucleus (order theory)

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

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

Every nucleus is evidently a monotone function.

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

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.^[2]

References

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