Noetherian induction
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 be such a set and let
be a subset of it having the property that for every
there is a strictly smaller element
. Then
is empty. For example, let
be the set of all closed subsets of a Noetherian space and let
be the set of those closed subsets that cannot be represented as a finite union of irreducible components. If
, then
is reducible, that is,
, where
and
are closed, both are strictly contained in
and at least one of them belongs to
. Consequently,
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.
Noetherian induction. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Noetherian_induction&oldid=12536