The Bourbaki-Witt theorem, named for Nicolas Bourbaki and Ernst Witt, is a fixed-point theorem for partially ordered sets. It states that if a nonempty poset is chain complete, meaning every chain within it has a least upper bound, and a function from the poset to itself never sends any element to something smaller than that element, then the function must have a fixed point, an element it leaves unchanged. Such a function is called inflationary or progressive, and the theorem is a foundational order-theoretic tool that underlies later fixed-point results used across set theory and computer science.
Facts
StatementIf X is a non-empty poset that is chain complete, meaning each chain has a least upper bound, and f from X to X is a function such that f(x) >= x for all x, then f has a fixed point. Such a function f is called inflationary or progressive. 2 Classification
Statement Form Connections
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Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
In Branch
Source Bourbaki-Witt theorem, Wikipedia
Sources
1. Wikipedia: Bourbaki-Witt theorem
WikipediaLead section, statement-form referenceQuote, Lead section, statement-form reference
It states that if X is a non-empty poset that is chain complete, meaning each chain has a least upper bound, and f : X → X is a function such that f ( x ) ≥ x for all x , then f has a fixed point.
2. Bourbaki-Witt theorem, Wikipedia
Lead paragraph, statement of the theorem
if X is a non-empty poset that is chain complete, meaning each chain has a least upper bound, and
- In Branch: Order Theory, Lead sentence
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