Nucleus (order theory) explained

In mathematics, and especially in order theory, a nucleus is a function

F on a meet-semilattice ak{A} such that (for every p in ak{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.

Frames and locales

Usually, the term nucleus is used in frames and locales theory (when the semilattice

ak{A} is a frame).

Proposition: If

F is a nucleus on a frame ak{A} , then the poset \operatorname{Fix}(F) of fixed points of F , with order inherited from ak{A} , is also a frame.^[1]

Notes and References

1. Book: Miraglia, Francisco. An Introduction to Partially Ordered Structures and Sheaves. Theorem 13.2, p. 130. Polimetrica s.a.s.. 2006. 9788876990359.