Nucleus (order theory)

In mathematics, and especially in order theory, a nucleus is a function  F {\displaystyle F} on a meet-semilattice A {\displaystyle {\mathfrak {A}}} such that (for every  p {\displaystyle p} in  A {\displaystyle {\mathfrak {A}}} ):[1]

1. p ≤ F ( p ) {\displaystyle p\leq F(p)}
2. F ( F ( p ) ) = F ( p ) {\displaystyle F(F(p))=F(p)}
3. F ( p ∧ q ) = F ( p ) ∧ F ( q ) {\displaystyle F(p\wedge q)=F(p)\wedge F(q)}

Every nucleus is evidently a monotone function.

## Frames and locales

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Usually, the term nucleus is used in frames and locales theory (when the semilattice  A {\displaystyle {\mathfrak {A}}} is a frame).

Proposition: If  F {\displaystyle F} is a nucleus on a frame  A {\displaystyle {\mathfrak {A}}} , then the poset Fix ⁡ F {\displaystyle \operatorname {Fix} F} of fixed points of  F {\displaystyle F} , with order inherited from  A {\displaystyle {\mathfrak {A}}} , is also a frame.[2]

## References

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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.