Difference between revisions of "Nucleus"
From Encyclopedia of Mathematics
VictorPorton (talk | contribs) (Created page with "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</...") |
VictorPorton (talk | contribs) (Every nucleus is evidently a monotone function.) |
||
| Line 4: | Line 4: | ||
# <math>F(F(p)) = F(p)</math> | # <math>F(F(p)) = F(p)</math> | ||
# <math>F(p \wedge q) = F(p) \wedge F(q)</math> | # <math>F(p \wedge q) = F(p) \wedge F(q)</math> | ||
| + | |||
| + | Every nucleus is evidently a monotone function. | ||
Usually, the term ''nucleus'' is used in [[frames and locales]] theory (when the semilattice <math>\mathfrak{A}</math> is a frame). | Usually, the term ''nucleus'' is used in [[frames and locales]] theory (when the semilattice <math>\mathfrak{A}</math> is a frame). | ||
Revision as of 15:28, 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=35679
Nucleus. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Nucleus&oldid=35679