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.