Infinite descending chain
- Infinite descending chain
Given a set "S" with a partial order ≤, an infinite descending chain is a chain "V", that is, a subset of "S" upon which ≤ defines a total order, such that "V" has no least element, that is, an element "m" such that for all elements "n" in "V" it holds that "m" ≤ "n".
As an example, in the set of integers, the chain −1, −2, −3, ... is an infinite descending chain, but there exists no infinite chain on the natural numbers, as every chain of natural numbers has a minimal element.
If a partially ordered set does not contain any infinite descending chains, it is called well-founded. A totally ordered set without infinite descending chains is called well-ordered.
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