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, 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 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.
In recursion theory, the mathematical theory of computability, a maximalset is a coinfinite recursively enumerable subset A of the natural numbers such...
graph theory, a maximal independent set (MIS) or maximal stable set is an independent set that is not a subset of any other independent set. In other words...
a faction of Transformers Maximalism, an artistic style Maximalset Maxim (magazine), a men's magazine marketed as Maximal in several countries Minimal...
theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all proper ideals. In other words, I is a maximal ideal of...
mathematics, especially in order theory, a maximal element of a subset S {\displaystyle S} of some preordered set is an element of S {\displaystyle S} that...
proving bounds on a circular maximal function analogous to the Kakeya maximal function. It was conjectured that there existed sets containing a sphere around...
any partially ordered set, every totally ordered subset is contained in a maximal totally ordered subset. The Hausdorff maximal principle is one of many...
In scale (music) theory, a maximally even set (scale) is one in which every generic interval has either one or two consecutive integers specific intervals-in...
strictly. In other words, H is a maximal element of the partially ordered set of subgroups of G that are not equal to G. Maximal subgroups are of interest because...
V̇O2 max (also maximal oxygen consumption, maximal oxygen uptake or maximal aerobic capacity) is the maximum rate of oxygen consumption attainable during...
} A maximal element of ( P , ≤ ) {\displaystyle (P,\leq )} is defined to mean a maximal element of the subset S := P . {\displaystyle S:=P.} A set can...
referred to as characterizing integrable systems: the existence of a maximalset of conserved quantities (the usual defining property of complete integrability)...
mind. Basic states are characterized by a set of quantum numbers. This is a set of eigenvalues of a maximalset of commuting observables. Physical observables...
isomorphic to the standard torus Tn). A maximal torus is one which is maximal among such subgroups. That is, T is a maximal torus if for any torus T′ containing...
to find a maximalset with certain properties. A binary relation R {\displaystyle R} on a set X {\displaystyle X} is formally defined as a set of ordered...
(Assumption) If G does not prove A, then we can construct an (infinite) maximalset, G∗, which is a superset of G and which also does not prove A. Place...
In computational geometry, a point p in a finite set of points S is said to be maximal or non-dominated if there is no other point q in S whose coordinates...
A ∈ S {\displaystyle A\in S} or B ∈ S {\displaystyle B\in S} . Maximal consistent sets are a fundamental tool in the model theory of classical logic and...
graphs while the independent set problem remains NP-hard on planar graphs. A maximal clique, sometimes called inclusion-maximal, is a clique that is not included...
commutative ring as the topology such that a set of maximal ideals is closed if and only if it is the set of all maximal ideals that contain a given ideal. Another...
partially ordered set does not even have any maximal elements, since any g divides for instance 2g, which is distinct from it, so g is not maximal. If the number...
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