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Difference between revisions of "Noetherian induction"

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A reasoning principle applicable to a [[Partially ordered set|partially ordered set]] in which every non-empty subset contains a minimal element; for example, the set of closed subsets in some [[Noetherian space|Noetherian space]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066810/n0668101.png" /> be such a set and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066810/n0668102.png" /> be a subset of it having the property that for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066810/n0668103.png" /> there is a strictly smaller element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066810/n0668104.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066810/n0668105.png" /> is empty. For example, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066810/n0668106.png" /> be the set of all closed subsets of a Noetherian space and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066810/n0668107.png" /> be the set of those closed subsets that cannot be represented as a finite union of irreducible components. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066810/n0668108.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066810/n0668109.png" /> is reducible, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066810/n06681010.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066810/n06681011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066810/n06681012.png" /> are closed, both are strictly contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066810/n06681013.png" /> and at least one of them belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066810/n06681014.png" />. Consequently, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066810/n06681015.png" /> is empty.
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A reasoning principle applicable to a [[Partially ordered set|partially ordered set]] in which every non-empty subset contains a minimal element; for example, the set of closed subsets in some [[Noetherian space|Noetherian space]]. Let $M$ be such a set and let $F$ be a subset of it having the property that for every $a\in F$ there is a strictly smaller element $b\in F$. Then $F$ is empty. For example, let $M$ be the set of all closed subsets of a Noetherian space and let $F$ be the set of those closed subsets that cannot be represented as a finite union of irreducible components. If $Y\in F$, then $Y$ is reducible, that is, $Y=Y_1\cup Y_2$, where $Y_1$ and $Y_2$ are closed, both are strictly contained in $Y$ and at least one of them belongs to $F$. Consequently, $F$ is empty.
  
 
Reversal of the order makes it possible to apply Noetherian induction to partially ordered sets in which every non-empty subset contains a maximal element; for example, to the lattice of ideals in a [[Noetherian ring|Noetherian ring]].
 
Reversal of the order makes it possible to apply Noetherian induction to partially ordered sets in which every non-empty subset contains a maximal element; for example, to the lattice of ideals in a [[Noetherian ring|Noetherian ring]].

Latest revision as of 08:34, 1 August 2014

A reasoning principle applicable to a partially ordered set in which every non-empty subset contains a minimal element; for example, the set of closed subsets in some Noetherian space. Let $M$ be such a set and let $F$ be a subset of it having the property that for every $a\in F$ there is a strictly smaller element $b\in F$. Then $F$ is empty. For example, let $M$ be the set of all closed subsets of a Noetherian space and let $F$ be the set of those closed subsets that cannot be represented as a finite union of irreducible components. If $Y\in F$, then $Y$ is reducible, that is, $Y=Y_1\cup Y_2$, where $Y_1$ and $Y_2$ are closed, both are strictly contained in $Y$ and at least one of them belongs to $F$. Consequently, $F$ is empty.

Reversal of the order makes it possible to apply Noetherian induction to partially ordered sets in which every non-empty subset contains a maximal element; for example, to the lattice of ideals in a Noetherian ring.

References

[1] N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)


Comments

The term well-founded induction is also in use.

How to Cite This Entry:
Noetherian induction. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Noetherian_induction&oldid=32633
This article was adapted from an original article by L.V. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article