Maximal set (computability theory): Difference between revisions

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In [[recursioncomputability theory]], the mathematical theory of computability, a '''maximal set''' is a coinfinite [[recursivelycomputably enumerable set|recursivelycomputably enumerable subset]] ''A'' of the [[natural number|natural numbers]]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.

{{wikify|date=July 2007}}

Maximal sets have many interesting properties: they are [[simple set|simple]], [[hypersimple set|hypersimple]], [[hyperhypersimple]]<!-- These terms should be defined and given context. --> and r-maximal; 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)<!-- I DON'T SEE A PAPER BY MYHILL CITED HERE. --> asked whether maximal sets existsexist and FriebergFriedberg (1958) constructed one. Soare (1974)<!-- SAME COMMENT HERE. --> 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 mathematical theory of computability, a '''maximal set''' is a coinfinite [[recursively enumerable set|recursively enumerable subset]] ''A'' of the [[natural number|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 (order)|lattice]] of the recursively enumerable sets.

Maximal sets have many interesting properties: they are [[simple set|simple]], [[hypersimple]], [[hyperhypersimple]]<!-- These terms should be defined and given context. --> 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)<!-- I DON'T SEE A PAPER BY MYHILL CITED HERE. --> asked whether maximal sets exists and Frieberg (1958) constructed one. Soare (1974)<!-- SAME COMMENT HERE. --> 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 |mr=0109125 | year=1958 | journal=The Journal of Symbolic Logic | volume=23 | pages=309–316 | doi=10.2307/2964290 | issue=3 | publisher=Association for Symbolic Logic | jstor=2964290| s2cid=25834814 }}

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

* {{Citation | last1=Myhill | first1=John | title=Solution of a problem of Tarski |mr=0075894 | year=1956 | journal=The Journal of Symbolic Logic | 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 |mr=0360235 | year=1974 | journal=[[Annals of Mathematics]] |series=Second 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]]