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Difference between revisions of "Nucleus"

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(→‎References: Johnstone (2002))
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In mathematics, and especially in [[order theory]], a '''nucleus''' is a function <math>F</math> on a [[meet-semilattice]] <math>\mathfrak{A}</math> such that (for every <math>p</math> in <math>\mathfrak{A}</math>):
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''on a [[partially ordered set]]''
  
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# <math>p \le F(p)</math>
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A function $F$ on a meet-[[semi-lattice]] $\mathfrak{A}$ such that (for every $p \in \mathfrak{A}$):
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# <math>F(F(p)) = F(p)</math>
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$$
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# <math>F(p \wedge q) = F(p) \wedge F(q)</math>
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p \le F(p)\ ;
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$$
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$$
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F(F(p)) = F(p)\ ;
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$$
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$$
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F(p \wedge q) = F(p) \wedge F(q) \ .
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$$
  
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Every nucleus is evidently a monotone function.
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Every nucleus is evidently a [[monotone function]].  A nucleus is determined by its [[poset]] $\operatorname{Fix}(F)$ of fixed points, since $F$ is left adjoint to the embedding $\operatorname{Fix}(F) \hookrightarrow \mathfrak{A}$.
  
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Usually, the term ''nucleus'' is used in [[frames and locales]] theory (when the semilattice <math>\mathfrak{A}</math> is a frame).
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Usually, the term ''nucleus'' is used in [[Locale|frames and locales]] theory (when the semilattice $\mathfrak{A}$ is a frame).  If $F$ is a nucleus on a frame $\mathfrak{A}$, then $\operatorname{Fix}(F)$ with order inherited from $\mathfrak{A}$ is also a frame.
  
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==Some well known results about nuclei==
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====References====
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<table>
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'''Proposition:''' If <math>F</math> is a nucleus on a frame <math>\mathfrak{A}</math>, then the poset <math>\operatorname{Fix}(F)</math> of fixed points of <math>F</math>, with order inherited from <math>\mathfrak{A}</math>, is also a frame.
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<TR><TD valign="top">[a1]</TD> <TD valign="top"> Peter T. Johnstone ''Sketches of an elephant'' Oxford University Press (2002) ISBN 0198534256 {{ZBL|1071.18001}}</TD></TR>
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</table>

Revision as of 21:20, 7 January 2016

on a partially ordered set

A function $F$ on a meet-semi-lattice $\mathfrak{A}$ such that (for every $p \in \mathfrak{A}$): $$ p \le F(p)\ ; $$ $$ F(F(p)) = F(p)\ ; $$ $$ F(p \wedge q) = F(p) \wedge F(q) \ . $$

Every nucleus is evidently a monotone function. A nucleus is determined by its poset $\operatorname{Fix}(F)$ of fixed points, since $F$ is left adjoint to the embedding $\operatorname{Fix}(F) \hookrightarrow \mathfrak{A}$.

Usually, the term nucleus is used in frames and locales theory (when the semilattice $\mathfrak{A}$ is a frame). If $F$ is a nucleus on a frame $\mathfrak{A}$, then $\operatorname{Fix}(F)$ with order inherited from $\mathfrak{A}$ is also a frame.

References

[a1] Peter T. Johnstone Sketches of an elephant Oxford University Press (2002) ISBN 0198534256 Zbl 1071.18001
How to Cite This Entry:
Nucleus. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Nucleus&oldid=37386
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