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Difference between revisions of "Nucleus"

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====References====
 
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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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<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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Latest revision as of 16:48, 23 November 2023

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