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In recursion theory, the mathematical theory of computability, a maximal set is a coinfinite recursively enumerable subset A of the natural numbers such that for every further recursively enumerable subset B of the natural numbers, either B is cofinite or B is a finite variant of A or B is not a superset of A. This gives an easy definition within the lattice of the recursively enumerable sets.

Attributes	Values
rdfs:label	Maximal set (en)
rdfs:comment	In recursion theory, the mathematical theory of computability, a maximal set is a coinfinite recursively enumerable subset A of the natural numbers such that for every further recursively enumerable subset B of the natural numbers, either B is cofinite or B is a finite variant of A or B is not a superset of A. This gives an easy definition within the lattice of the recursively enumerable sets. (en)
dct:subject	Computability theory
Wikipage page ID	10495079 (xsd:integer)
Wikipage revision ID	814475074 (xsd:integer)
Link from a Wikipage to another Wikipage	Annals of Mathematics Hypersimple set Computability Mathematics Modulo (jargon) Automorphism Lattice (order) Computability theory Natural number Recursively enumerable set Simple set Recursion theory Cofinite dbr:Hyperhypersimple
sameAs	Maximal set Maximal set Maximal set
dbp:wikiPageUsesTemplate	dbt:Citation dbt:Isbn dbt:Mathlogic-stub
has abstract	In recursion theory, the mathematical theory of computability, a maximal set is a coinfinite recursively enumerable subset A of the natural numbers such that for every further recursively enumerable subset B of the natural numbers, either B is cofinite or B is a finite variant of A or B is not a superset of A. This gives an easy definition within the lattice of the recursively enumerable sets. Maximal sets have many interesting properties: they are simple, hypersimple, and r-maximal; the latter property says that every recursive set R contains either only finitely many elements of the complement of A or almost all elements of the complement of A. There are r-maximal sets that are not maximal; some of them do even not have maximal supersets. Myhill (1956) asked whether maximal sets exist and Friedberg (1958) constructed one. Soare (1974) showed that the maximal sets form an orbit with respect to automorphism of the recursively enumerable sets under inclusion (modulo finite sets). On the one hand, every automorphism maps a maximal set A to another maximal set B; on the other hand, for every two maximal sets A, B there is an automorphism of the recursively enumerable sets such that A is mapped to B. (en)
prov:wasDerivedFrom	wikipedia-en:Maximal_set?oldid=814475074&ns=0
page length (characters) of wiki page	2659 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Maximal_set
is Link from a Wikipage to another Wikipage of	Computability theory Anaphora (linguistics) Glossary of topology Gödel's incompleteness theorems Maximal sets Maximal
is Wikipage redirect of	Maximal sets
is Wikipage disambiguates of	Maximal
is foaf:primaryTopic of	wikipedia-en:Maximal_set

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