maximal set

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In
recursion theory Computability theory, also known as recursion theory, is a branch of mathematical logic, computer science, and the theory of computation that originated in the 1930s with the study of computable functions and Turing degrees. The field has since ...
, the
mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
theory of
computability Computability is the ability to solve a problem in an effective manner. It is a key topic of the field of computability theory within mathematical logic and the theory of computation within computer science. The computability of a problem is cl ...
, a maximal set is a coinfinite recursively enumerable subset ''A'' of the
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal ...
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 of the recursively enumerable sets. Maximal sets have many interesting properties: they are simple, hypersimple, hyperhypersimple 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 In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all 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

* * * H. Rogers, Jr., 1967. ''The Theory of Recursive Functions and Effective Computability'', second edition 1987, MIT Press. (paperback), . *
Computability theory {{mathlogic-stub