Automorphisms of the Lattice of 0 1 Classes; Perfect Thin Classes and Anc Degrees
نویسندگان
چکیده
0 1 classes are important to the logical analysis of many parts of mathematics. The 0 1 classes form a lattice. As with the lattice of computable enumerable sets, it is natural to explore the relationship between this lattice and the Turing degrees. We focus on an analog of maximality namely the notion of a thin class. We prove a number of results relating automorphisms, invariance and thin classes. Our main result is an analog of the Martin-Soare work on maximal sets and high degrees, using thin classes and anc degrees. In particular, we show that the perfect thin classes are deenable (in the lattice of 0 1 classes) and a degree is anc ii it contains a perfect thin class. Hence the class of anc degrees is an invariant class for the lattice of 0 1 classes. We show that all perfect thin classes are automorphic (via a 0 3 automorphism).
منابع مشابه
Automorphisms of the Lattice of Π1 Classes; Perfect Thin Classes and Anc Degrees
Π1 classes are important to the logical analysis of many parts of mathematics. The Π1 classes form a lattice. As with the lattice of computably enumerable sets, it is natural to explore the relationship between this lattice and the Turing degrees. We focus on an analog of maximality, or more precisely, hyperhypersimplicity, namely the notion of a thin class. We prove a number of results relatin...
متن کاملNILPOTENCY AND SOLUBILITY OF GROUPS RELATIVE TO AN AUTOMORPHISM
In this paper we introduce the concept of α-commutator which its definition is based on generalized conjugate classes. With this notion, α-nilpotent groups, α-solvable groups, nilpotency and solvability of groups related to the automorphism are defined. N(G) and S(G) are the set of all nilpotency classes and the set of all solvability classes for the group G with respect to different automorphi...
متن کاملThe upward closure of a perfect thin class
There is a perfect thin Π1 class whose upward closure in the Turing degrees has full measure (and indeed contains every 2-random degree.) Thus, in the Muchnik lattice of Π1 classes, the degree of 2-random reals is comparable with the degree of some perfect thin class. This solves a question of Simpson [15].
متن کاملImmunity and Non-Cupping for Closed Sets
We extend the notion of immunity to closed sets and to Π1 classes in particular in two ways: immunity meaning the corresponding tree has no infinite computable subset, and tree-immunity meaning it has no infinite computable subtree. We separate these notions from each other and that of being special, and show separating classes for computably inseparable c.e. sets are immune and perfect thin cl...
متن کاملSome connections between powers of conjugacy classes and degrees of irreducible characters in solvable groups
Let $G$ be a finite group. We say that the derived covering number of $G$ is finite if and only if there exists a positive integer $n$ such that $C^n=G'$ for all non-central conjugacy classes $C$ of $G$. In this paper we characterize solvable groups $G$ in which the derived covering number is finite.
متن کامل