Nucleus

In order theory, a nucleus is a function $F$ on a meet-semilattice $\mathfrak{A}$ such that (for every $p$ in $\mathfrak{A}$ ):

1.

$p\leq F(p)$
2.

$F(F(p))=F(p)$
3.

$F(p\wedge q)=F(p)\wedge F(q)$

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

1 Some well known results about nuclei

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.

Title	Nucleus
Canonical name	Nucleus
Date of creation	2014-12-18 15:34:14
Last modified on	2014-12-18 15:34:14
Owner	porton (9363)
Last modified by	porton (9363)
Numerical id	1
Author	porton (9363)
Entry type	Definition
Classification	msc 06B99