12/08/14 07:14:47.33
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Ascending and descending chain conditions
In this epoch, Noether became famous for her deft use of ascending (Teilerkettensatz) or descending (Vielfachenkettensatz) chain conditions.
A sequence of non-empty subsets A1, A2, A3, etc. of a set S is usually said to be ascending, if each is a subset of the next
A1⊂A2⊂A3⊂・・・
Conversely, a sequence of subsets of S is called descending if each contains the next subset:
A1⊃A2⊃A3⊃・・・
A chain becomes constant after a finite number of steps if there is an n such that for all m ? n.
A collection of subsets of a given set satisfies the ascending chain condition if any ascending sequence becomes constant after a finite number of steps.
It satisfies the descending chain condition if any descending sequence becomes constant after a finite number of steps.
Ascending and descending chain conditions are general, meaning that they can be applied to many types of mathematical objects?and, on the surface, they might not seem very powerful.
Noether showed how to exploit such conditions, however, to maximum advantage: for example, how to use them to show that every set of sub-objects has a maximal/minimal element or that a complex object can be generated by a smaller number of elements.
These conclusions often are crucial steps in a proof.
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