Determining an atmic Boolean algebra from the action of the automorphism group

1993 ◽  
Vol 34 (6) ◽  
pp. 1041-1043 ◽  
Author(s):  
B. K. Dauletbaev
Keyword(s):  
Automorphism Group ◽  
Boolean Algebra

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

1986 ◽  
Vol 51 (2) ◽  
pp. 292 ◽  
Author(s):  
E. W. Madison ◽  
B. Zimmermann-Huisgen
Keyword(s):  
Automorphism Group ◽  
Boolean Algebra ◽  
Atomless Boolean Algebra

Combinatorial and recursive aspects of the automorphism group of the countable atomless Boolean algebra

1986 ◽  
Vol 51 (2) ◽  
pp. 292-301
Author(s):  
E. W. Madison ◽  
B. Zimmermann-Huisgen
Keyword(s):  
Automorphism Group ◽  
Boolean Algebra ◽  
Recursive Function ◽  
Conjugacy Classes ◽  
Explicit Description ◽  
Free Basis ◽  
Atomless Boolean Algebra ◽  
Cycle Structures ◽  
Countably Infinite

AbstractGiven an admissible indexing φ of the countable atomless Boolean algebra ℬ, an automorphism F of ℬ is said to be recursively presented (relative to φ) if there exists a recursive function p ϵ Sym(ω) such that F ∘ φ = φ ∘ p. Our key result on recursiveness: Both the subset of Aut(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 ƒ and , where ƒ is a permutation of some free basis of ℬ and is the automorphism of ℬ induced by ƒ.(2) An explicit description of the permutations of ω whose conjugacy classes in Sym(ω) are (a) uncountable, (b) countably infinite, and (c) finite.

scholarly journals Universal minimal flows of automorphism groups

2003 ◽  
Vol 127 (28) ◽  
pp. 93-106
Author(s):  
A.S. Kechris ◽  
V. Pestov ◽  
S. Todorcevic
Keyword(s):  
Automorphism Group ◽  
Boolean Algebra ◽  
Automorphism Groups ◽  
Ramsey Theory ◽  
Topological Dynamics ◽  
Dimensional Vector ◽  
Topological Groups ◽  
Free Graph ◽  
Dimensional Vector Space ◽  
Atomless Boolean Algebra

We investigate some connections between the Fraiss? theory of amalgamation classes and ultrahomogeneous structures, Ramsey theory, and topological dynamics of automorphism groups of countable structures. We show, in particular, that results from the structural Ramsey theory can be quite useful in recognizing the universal minimal flows of this kind of groups. As result we compute universal minimal flows of several well known topological groups such as, for example, the automorphism group of the random graph, the automorphism group of the random triangle-free graph, the automorphism group of the ?-dimensional vector space over a finite field, the automorphism group of the countable atomless Boolean algebra, etc. So we have here a reversal in the traditional relationship between topological dynamics and Ramsey theory, the Ramsey-theoretic results are used in proving theorems of topological dynamics rather than vice versa.

Boolean algebras and orbits of the lattice of r.e sets modulo the finite sets

1990 ◽  
Vol 55 (2) ◽  
pp. 744-760
Author(s):  
Peter Cholak
Keyword(s):  
Automorphism Group ◽  
Boolean Algebra ◽  
Boolean Algebras ◽  
Finite Sets ◽  
Total Function ◽  
Class Of Functions

An important program in the study of the structure the lattice of r.e. sets modulo finite sets, is the classification of the orbits under Aut(), the automorphism group of , If Φ ∈ Aut() and Φ(We) =* Wh(e) for all e ∈ ω, then h is called a presentation of Φ. Define Autχ() to be the class of those elements of Aut() that have a presentation in the class of functions X, where for instance X might be the class of Δn functions for n ∈ ω. If Φ ∈ Autx() then we will say that Φ is an X-automorphism. Note that we only need to consider Δn- and Πn-orbits and automorphisms since if f is a Σn-presentation, then f is a total function and therefore Δn.Definition. If X is a class of functions and ⊆ {Wi: i < ω}, then is an X-orbit iff is an orbit under Aut() and for all A, B ∈ there is a Φ ∈ Autx() satisfying Φ(A) =* B. (Here Φ: → .)Harrington proved that the creative sets form a Δ0-orbit and Soare proved that the maximal sets form Δ3-orbit. We will show that there is no Boolean algebra such that {A: A is r.e. and ℒ*(A) ≈ ] forms a Δ2-orbit, where ℒ*(A) is the principal filter of A in ; ℒ*(A) = {B: B ⊆* A & B ∈ }. The idea behind the proof of this theorem is very similar to the proof by Soare that maximal sets do not form a Δ2-orbit (see Soare [1974] or Soare [1987]).

scholarly journals The uncountable cofinality of the automorphism group of the countable universal distributive lattice

2011 ◽  
Vol 44 (3) ◽  
Author(s):  
M. Droste ◽  
J. K. Truss
Keyword(s):  
Automorphism Group ◽  
Boolean Algebra ◽  
Distributive Lattice ◽  
Generalized Boolean Algebra

AbstractWe show that the automorphism group of the countable universal distributive lattice has strong uncountable cofinality, and we adapt the method to deduce the strong uncountable cofinality of the automorphism group of the countable universal generalized boolean algebra.

A consistent very small Boolean algebra with countable automorphism group

1980 ◽  
Vol 11 (1) ◽  
pp. 389-392 ◽  
Author(s):  
Eric K. van Douwen
Keyword(s):  
Automorphism Group ◽  
Boolean Algebra

Automorphism Groups of Denumerable Boolean Algebras

1977 ◽  
Vol 29 (3) ◽  
pp. 466-471 ◽  
Author(s):  
Ralph Mckenzie
Keyword(s):  
Automorphism Group ◽  
Boolean Algebra ◽  
Automorphism Groups ◽  
Boolean Algebras ◽  
Group Of Automorphisms

We are concerned with the extent to which the structure of a Boolean algebra (or BA, for brevity) is reflected in its group of automorphisms, Aut . In particular, for which algebras can one conclude that if Aut Aut , then Monk has conjectured [3] that this implication holds for denumerable BA's with at least one atom. We shall refute his conjecture, but show that the implication does hold if and are denumerable, if each has at least one atom, and if the sum of the atoms exists in . In fact, under those assumptions the algebra 21 can be rather neatly recovered from its abstract automorphism group.

scholarly journals Forcing a Boolean algebra with predesigned automorphism group

2002 ◽  
Vol 130 (10) ◽  
pp. 2837-2843
Author(s):  
Tapani Hyttinen ◽  
Saharon Shelah
Keyword(s):  
Automorphism Group ◽  
Boolean Algebra

ON THE AUTOMORPHISM GROUP OF A FINITE P-GROUP WITH CYCLIC FRATTINI SUBGROUP

2008 ◽  
Vol 108 (2) ◽  
pp. 165-175 ◽  
Author(s):  
S. Fouladi ◽  
A. R. Jamali ◽  
R. Orfi
Keyword(s):  
Automorphism Group ◽  
Frattini Subgroup

Synthesis of Antennas on the Base of the Boolean Algebra Transformations and the Atomic Functions Theory

2005 ◽  
Vol 64 (9) ◽  
pp. 699-712
Author(s):  
Victor Filippovich Kravchenko ◽  
Miklhail Alekseevich Basarab
Keyword(s):  
Boolean Algebra
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