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 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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