Nucleus (order Theory)

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

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

Every nucleus is evidently a monotone function.

Frames and locales

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

Proposition: If $F$ is a nucleus on a frame $\mathfrak$, then the poset $\operatorname(F)$ of fixed points of $F$, with order inherited from $\mathfrak$, is also a frame.

References

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Order theory