In mathematics a well-founded set (also well-founded set , well-founded order , terminating order , Noetherian order ) is a semi-ordered set that does not contain any infinite, genuinely descending chains. Equivalently, a semi-ordered set is called well-founded if every non-empty subset contains at least one minimal element .
All well-ordered sets are well-founded because in a well-ordered set every non-empty subset must have a smallest element and the smallest element of a set is always minimal. Unlike well-ordered sets, well-founded sets do not need to be totally ordered . All totally ordered well-founded sets are well ordered.
Profound amounts allow the application of noetherian induction, a version of transfinite induction : Is a property of elements of a under an order relation depth amount , and the following statements are true:
- is true for all minimal elements of .
- If an element of and is true for all , then is also true.
Then true for all elements of .
The whole numbers , the rational numbers, and the real numbers in their natural order each contain infinite descending chains and are therefore not well founded.
The power set of a set with the subset relation as order is well founded if and only if the set is finite . All finite semi-ordered sets are well-founded because finite sets can only have finite chains.
The following sets are well-founded but not totally ordered:
- the natural numbers with the order
- If a divisor of is
- the set of sub-modules of a noetheric module with the order
- the set of all pairs of natural numbers with order
- if and
- If a substring of is
- the set of regular expressions over a given alphabet with the order
- If a part of expression is
- loads of sets with the order
- , if is an element of (really element, not subset!)
Length of descending chains
If and is a well-founded set , then the descending chains at the beginning are all finite, but their length does not have to be restricted. Consider e.g. B. the amount
(where ) with the order
- if or
It is z. B. and . is well-founded, but there are descending chains of arbitrary (finite) length at the beginning.