Difference between revisions of "Nucleus"
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VictorPorton (talk | contribs) (Every nucleus is evidently a monotone function.) |
VictorPorton (talk | contribs) m (meet-semilattice link added) |
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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>): | + | 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>): |
# <math>p \le F(p)</math> | # <math>p \le F(p)</math> | ||
Revision as of 18:05, 18 December 2014
In mathematics, and especially in order theory, a nucleus is a function \(F\) on a meet-semilattice \(\mathfrak{A}\) 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.
Usually, the term nucleus is used in frames and locales theory (when the semilattice \(\mathfrak{A}\) is a frame).
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.
How to Cite This Entry:
Nucleus. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Nucleus&oldid=35680
Nucleus. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Nucleus&oldid=35680