Nucleus

on a partially ordered set

A function $F$ on a meet-semi-lattice $\mathfrak{A}$ such that (for every $p \in \mathfrak{A}$): $$ p \le F(p)\ ; $$ $$ F(F(p)) = F(p)\ ; $$ $$ F(p \wedge q) = F(p) \wedge F(q) \ . $$

Every nucleus is evidently a monotone function. A nucleus is determined by its poset $\operatorname{Fix}(F)$ of fixed points, since $F$ is left adjoint to the embedding $\operatorname{Fix}(F) \hookrightarrow \mathfrak{A}$.

Usually, the term nucleus is used in frames and locales theory (when the semilattice $\mathfrak{A}$ is a frame). If $F$ is a nucleus on a frame $\mathfrak{A}$, then $\operatorname{Fix}(F)$ with order inherited from $\mathfrak{A}$ is also a frame.

References

[a1]	Peter T. Johnstone Sketches of an elephant Oxford University Press (2002) ISBN 0198534256 Zbl 1071.18001