Nucleus

In order theory, a nucleus is afunction $F$ 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 thesemilattice $\\mathfrak{A}$ 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.

单词	Nucleus
释义	Nucleus In order theory, a nucleus is afunction $F$ 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 thesemilattice $\\mathfrak{A}$ 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.
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