Nucleus (order theory)

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]

1. $p \le F(p)$
2. $F(F(p)) = F(p)$
3. $F(p \wedge q) = F(p) \wedge F(q)$

Every nucleus is evidently a monotone function.

Frames and locales

Usually, the term nucleus is used in frames and locales theory (when the semilattice $\mathfrak{A}$ 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

1. Johnstone, Peter (1982), Stone Spaces, Cambridge University Press, p. 48, ISBN 978-0-521-33779-3, Zbl 0499.54001
2. Miraglia, Francisco (2006). An Introduction to Partially Ordered Structures and Sheaves. Polimetrica s.a.s. Theorem 13.2, p. 130. ISBN 9788876990359.