Maximal set (computability theory): Difference between revisions

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In [[recursioncomputability theory]], the [[mathematics|mathematical]] theory of computability, a '''maximal set''' is a coinfinite [[recursivelycomputably enumerable set|recursivelycomputably enumerable subset]] ''A'' of the [[natural number]]s such that for every further recursivelycomputably 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 (order)|lattice]] of the recursivelycomputably enumerable sets.

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Maximal sets have many interesting properties: they are [[simple set|simple]], [[hypersimple set|hypersimple]], [[hyperhypersimple]] and r-maximal;{{clarifyme}} the latter property says that every recursivecomputable 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 existsexist and Friedberg (1958) constructed one. Soare (1974) showed that the maximal sets form an orbit with respect to [[automorphism]] of the recursivelycomputably enumerable sets under inclusion ([[Modulo (jargon)|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 recursivelycomputably enumerable sets such that ''A'' is mapped to ''B''.

In [[recursion theory]], the [[mathematics|mathematical]] theory of computability, a '''maximal set''' is a coinfinite [[recursively enumerable set|recursively enumerable subset]] ''A'' of the [[natural number]]s 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 (order)|lattice]] of the recursively enumerable sets.

Maximal sets have many interesting properties: they are [[simple set|simple]], [[hypersimple]], [[hyperhypersimple]] and r-maximal;{{clarifyme}} 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 exists 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''.

==References==

* {{Citation | last1=Friedberg | first1=Richard M. | title=Three theorems on recursive enumeration. I. Decomposition. II. Maximal set. III. Enumeration without duplication | idmr={{MathSciNet | id = 0109125}} | year=1958 | journal=The Journal of Symbolic Logic | issn=0022-4812 | volume=23 | pages=309–316 | doi=10.2307/2964290 | issue=3 | publisher=Association for Symbolic Logic | jstor=2964290| s2cid=25834814 }}

* {{Citation | last1=Myhill | first1=John | title=Solution of a problem of Tarski | idmr={{MathSciNet | id = 0075894}} | year=1956 | journal=The Journal of Symbolic Logic | issn=0022-4812 | volume=21 | pages=49–51 | doi=10.2307/2268485 | issue=1 | publisher=Association for Symbolic Logic | jstor=2268485| s2cid=19695459 }}

* H. Rogers, Jr., 1967. ''The Theory of Recursive Functions and Effective Computability'', second edition 1987, MIT Press. ISBN {{isbn|0-262-68052-1}} (paperback), ISBN {{isbn|0-07-053522-1}}.

* {{Citation | last1=Soare | first1=Robert I. | title=Automorphisms of the lattice of recursively enumerable sets. I. Maximal sets | doi=10.2307/1970842 | id={{MathSciNet | id mr= 0360235}} | year=1974 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issnseries=0003-486XSecond Series | volume=100 | pages=80–120 | issue=1 | publisher=Annals of Mathematics | jstor=1970842}}

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[[Category:RecursionComputability theory]]

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[[Category:Recursion theory]]

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