Journal of Symbolic Logic

Combinatorial and Recursive Aspects of the Automorphism Group of the Countable Atomless Boolean Algebra

Abstract

Given an admissible indexing $\varphi$ of the countable atomless Boolean algebra $\mathscr{B}$, an automorphism $F$ of $\mathscr{B}$ is said to be recursively presented (relative to $\varphi$) if there exists a recursive function $p \in \operatorname{Sym}(\omega)$ such that $F \circ \varphi = \varphi \circ p$. Our key result on recursiveness: Both the subset of $\operatorname{Aut}(\mathscr{B})$ consisting of all those automorphisms which are recursively presented relative to some indexing, and its complement, the set of all "totally nonrecursive" automorphisms, are uncountable. This arises as a consequence of the following combinatorial investigations: (1) A comparison of the cycle structures of $f$ and $\bar{f}$, where $f$ is a permutation of some free basis of $\mathscr{B}$ and $\bar{f}$ is the automorphism of $\mathscr{B}$ induced by $f$. (2) An explicit description of the permutations of $\omega$ whose conjugacy classes in $\operatorname{Sym}(\omega)$ are (a) uncountable, (b) countably infinite, and (c) finite.

Article information

Source
J. Symbolic Logic, Volume 51, Issue 2 (1986), 292-301.

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183742094

Mathematical Reviews number (MathSciNet)
MR840406

Zentralblatt MATH identifier
0621.03031

JSTOR