Nucleus (order theory)
In mathematics, and especially in order theory, a nucleus is a function [math]\displaystyle{ F }[/math] on a meet-semilattice [math]\displaystyle{ \mathfrak{A} }[/math] such that (for every [math]\displaystyle{ p }[/math] in [math]\displaystyle{ \mathfrak{A} }[/math]):[1]
- [math]\displaystyle{ p \le F(p) }[/math]
- [math]\displaystyle{ F(F(p)) = F(p) }[/math]
- [math]\displaystyle{ F(p \wedge q) = F(p) \wedge F(q) }[/math]
Every nucleus is evidently a monotone function.
Frames and locales
Usually, the term nucleus is used in frames and locales theory (when the semilattice [math]\displaystyle{ \mathfrak{A} }[/math] is a frame).
Proposition: If [math]\displaystyle{ F }[/math] is a nucleus on a frame [math]\displaystyle{ \mathfrak{A} }[/math], then the poset [math]\displaystyle{ \operatorname{Fix}(F) }[/math] of fixed points of [math]\displaystyle{ F }[/math], with order inherited from [math]\displaystyle{ \mathfrak{A} }[/math], is also a frame.[2]
References
- ↑ Johnstone, Peter (1982), Stone Spaces, Cambridge University Press, p. 48, ISBN 978-0-521-33779-3, https://books.google.com/books?id=CiWwoLNbpykC&pg=PA48
- ↑ Miraglia, Francisco (2006). An Introduction to Partially Ordered Structures and Sheaves. Polimetrica s.a.s.. Theorem 13.2, p. 130. ISBN 9788876990359. https://books.google.com/books?id=w6or65-Hw6UC&pg=PA130.
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