Pataraia's theorem, introduced by Dito Pataraia in 1997, states that every monotone map from an inductive poset to itself has a least fixed point, where an inductive poset is a directed complete partial order that has a least element. The result is a variant of the earlier Bourbaki-Witt fixed-point theorem, sharing its conclusion that a well-behaved monotone self-map of an appropriate ordered structure must fix some point, and it is used in order theory and in the domain-theoretic foundations of programming language semantics wherever a least fixed point of a monotone operator is needed.
Facts
StatementEvery monotone map from an inductive poset to itself admits a least fixed point, where an inductive poset is a directed complete partial order that has a least element. 1 Classification
Statement Form Connections
Has Statement Form
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.
Sources
1. Pataraia's theorem, Wikipedia
Lede section, first sentence
In mathematics, Pataraia's theorem states each monotone map f : P to P for an inductive poset P admits a least fixed point, where an inductive poset means a dcpo with a least element.
Lede section, second sentence
It was introduced by Dito Pataraia in 1997.
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