Embedding theorems for boolean algebras and consistency results on ordinal definable sets

1977 ◽  
Vol 42 (1) ◽  
pp. 64-76 ◽  
Author(s):  
Petr Štěpánek ◽  
Bohuslav Balcar
Keyword(s):  
Boolean Algebra ◽  
Extreme Case ◽  
Chain Condition ◽  
Boolean Algebras ◽  
Complete Boolean Algebra ◽  
Suslin Tree ◽  
Consistency Results ◽  
Weak Homogeneity ◽  
Uncountable Cardinal ◽  
Definable Sets

The existence of complete rigid Boolean algebras was first proved by McAloon [8] who also showed that every Boolean algebra can be completely embedded in a rigid complete Boolean algebra. McAloon was interested in consistency results on ordinal definable sets. His approach was based on forcing. Recently, Shelah [10] proved that for every uncountable cardinal κ there exists a Boolean algebra of power κ with rigid completion. Extending his method, we get the following theorems.Theorem 1. Any Boolean algebra B can be completely embedded in a complete Boolean algebra C with no nontrivial σ-complete one-one endomor-phism. If B satisfies the κ-chain condition for an uncountable cardinal κ, the same holds true for C.Since every automorphism is a complete endomorphism, it follows from Theorem 1 that C is rigid. The other extreme case of Boolean algebras are homogeneous algebras. It was proved by Kripke [7] that every Boolean algebra can be completely embedded in a homogeneous complete Boolean algebra. In his proof, the homogeneous algebra contains antichains of cardinality equal to the power of the embedded Boolean algebra. The following result shows that this is essential: the analogue of Theorem 1 is not provable in set theory even for Boolean algebras with a very weak homogeneity property. We use a Suslin tree with particular properties constructed by Jensen [6] in conjunction with a forcing argument.

scholarly journals Boolean algebras of projections in (DF)-and (LF)-spaces

2003 ◽  
Vol 67 (2) ◽  
pp. 297-303 ◽  
Author(s):  
J. Bonet ◽  
W. J. Ricker
Keyword(s):  
Boolean Algebra ◽  
Spectral Measure ◽  
Boolean Algebras ◽  
Complete Boolean Algebra ◽  
Locally Convex Spaces ◽  
Important Class ◽  
Locally Convex ◽  
Convex Spaces

Conditions are presented which ensure that an abstractly σ-complete Boolean algebra of projections on a (DF)-space or on an (LF)-space is necessarily equicontinuous and/or the range of a spectral measure. This is an extension, to a large and important class of locally convex spaces, of similar and well known results due to W. Bade (respectively, B. Walsh) in the setting of normed (respectively metrisable) spaces.

scholarly journals Four games on Boolean algebras

Filomat ◽  
2016 ◽  
Vol 30 (13) ◽  
pp. 3389-3395
Author(s):  
Milos Kurilic ◽  
Boris Sobot
Keyword(s):  
Boolean Algebra ◽  
Winning Strategy ◽  
Zero Element ◽  
Boolean Algebras ◽  
The Other ◽  
Complete Boolean Algebra ◽  
Other Hand

The games G2 and G3 are played on a complete Boolean algebra B in ?-many moves. At the beginning White picks a non-zero element p of B and, in the n-th move, White picks a positive pn < p and Black chooses an in ? {0,1}. White wins G2 iff lim inf pin,n = 0 and wins G3 iff W A?[?]? ? n?A pin,n = 0. It is shown that White has a winning strategy in the game G2 iff White has a winning strategy in the cut-and-choose game Gc&c introduced by Jech. Also, White has a winning strategy in the game G3 iff forcing by B produces a subset R of the tree <?2 containing either ??0 or ??1, for each ? ? <?2, and having unsupported intersection with each branch of the tree <?2 belonging to V. On the other hand, if forcing by B produces independent (splitting) reals then White has a winning strategy in the game G3 played on B. It is shown that ? implies the existence of an algebra on which these games are undetermined.

On the strength of the Sikorski extension theorem for Boolean algebras

1983 ◽  
Vol 48 (3) ◽  
pp. 841-846 ◽  
Author(s):  
J.L. Bell
Keyword(s):  
Boolean Algebra ◽  
Prime Ideal ◽  
Extension Theorem ◽  
Boolean Algebras ◽  
Complete Boolean Algebra ◽  
The Axiom Of Choice ◽  
Universe Of Sets ◽  
The Universe ◽  
Boolean Extension ◽  
Prime Ideal Theorem

The Sikorski Extension Theorem [6] states that, for any Boolean algebra A and any complete Boolean algebra B, any homomorphism of a subalgebra of A into B can be extended to the whole of A. That is,Inj: Any complete Boolean algebra is injective (in the category of Boolean algebras).The proof of Inj uses the axiom of choice (AC); thus the implication AC → Inj can be proved in Zermelo-Fraenkel set theory (ZF). On the other hand, the Boolean prime ideal theoremBPI: Every Boolean algebra contains a prime ideal (or, equivalently, an ultrafilter)may be equivalently stated as:The two element Boolean algebra 2 is injective,and so the implication Inj → BPI can be proved in ZF.In [3], Luxemburg surmises that this last implication cannot be reversed in ZF. It is the main purpose of this paper to show that this surmise is correct. We shall do this by showing that Inj implies that BPI holds in every Boolean extension of the universe of sets, and then invoking a recent result of Monro [5] to the effect that BPI does not yield this conclusion.

scholarly journals Boolean algebras of projections

1975 ◽  
Vol 19 (3) ◽  
pp. 287-289
Author(s):  
P. G. Spain
Keyword(s):  
Hilbert Space ◽  
Boolean Algebra ◽  
Von Neumann Algebra ◽  
Boolean Algebras ◽  
Complete Boolean Algebra ◽  
Von Neumann ◽  
Relative Compactness ◽  
Weakly Closed ◽  
Result Theorem ◽  
Bounded Sets

Bade, in (1), studied Boolean algebras of projections on Banach spaces and showed that a σ-complete Boolean algebra of projections on a Banach space enjoys properties formally similar to those of a Boolean algebra of projections on Hilbert space. (His exposition is reproduced in (7: XVII).) Edwards and Ionescu Tulcea showed that the weakly closed algebra generated by a σ-complete Boolean algebra of projections can be represented as a von Neumann algebra; and that the representation isomorphism can be chosen to be norm, weakly, and strongly bicontinuous on bounded sets (8): Bade's results were then seen to follow from their Hilbert space counterparts. I show here that it is natural to relax the condition of σ-completeness to weak relative compactness; indeed, a Boolean algebra of projections has σ-completion if and only if it is weakly relatively compact (Theorem 1). Then, following the derivation of the theorem of Edwards and Ionescu Tulcea from the Vidav characterisation of abstract C*-algebras (see (9)), I give a result (Theorem 2) which, with its corollary, includes (1: 2.7, 2.8, 2.9, 2.10, 3.2, 3.3, 4.5).

scholarly journals On countably closed complete Boolean algebras

1996 ◽  
Vol 61 (4) ◽  
pp. 1380-1386
Author(s):  
Thomas Jech ◽  
Saharon Shelah
Keyword(s):  
Boolean Algebra ◽  
Boolean Algebras ◽  
Complete Boolean Algebra

AbstractIt is unprovable that every complete subalgebra of a countably closed complete Boolean algebra is countably closed.

Cardinal collapsing and ordinal definability

1978 ◽  
Vol 43 (4) ◽  
pp. 635-642 ◽  
Author(s):  
Petr Štěpánek
Keyword(s):  
Boolean Algebra ◽  
Set Theory ◽  
Axiom Of Choice ◽  
Complete Boolean Algebra ◽  
Ground Model ◽  
Inner Model ◽  
Definable Sets ◽  
The Axiom Of Choice ◽  
Generic Extensions ◽  
Models Of Set Theory

We shall describe Boolean extensions of models of set theory with the axiom of choice in which cardinals are collapsed by mappings definable from parameters in the ground model. In particular, starting from the constructible universe, we get Boolean extensions in which constructible cardinals are collapsed by ordinal definable sets.Let be a transitive model of set theory with the axiom of choice. Definability of sets in the generic extensions of is closely related to the automorphisms of the corresponding Boolean algebra. In particular, if G is an -generic ultrafilter on a rigid complete Boolean algebra C, then every set in [G] is definable from parameters in . Hence if B is a complete Boolean algebra containing a set of forcing conditions to collapse some cardinals in , it suffices to construct a rigid complete Boolean algebra C, in which B is completely embedded. If G is as above, then [G] satisfies “every set is -definable” and the inner model [G ∩ B] contains the collapsing mapping determined by B. To complete the result, it is necessary to give some conditions under which every cardinal from [G ∩ B] remains a cardinal in [G].The absolutness is granted for every cardinal at least as large as the saturation of C. To keep the upper cardinals absolute, it often suffices to construct C with the same saturation as B. It was shown in [6] that this is always possible, namely, that every Boolean algebra can be completely embedded in a rigid complete Boolean algebra with the same saturation.

scholarly journals The left, the right and the sequential topology on Boolean algebras

Filomat ◽  
2019 ◽  
Vol 33 (14) ◽  
pp. 4451-4459
Author(s):  
Milos Kurilic ◽  
Aleksandar Pavlovic
Keyword(s):  
Boolean Algebra ◽  
A Priori ◽  
Boolean Algebras ◽  
The Other ◽  
Complete Boolean Algebra ◽  
A Posteriori ◽  
Other Hand ◽  
Convergence Of Sequences ◽  
Algebraic Convergence ◽  
The Right

For the algebraic convergence ?s, which generates the well known sequential topology ?s on a complete Boolean algebra B, we have ?s = ?ls ? ?li, where the convergences ?ls and ?li are defined by ?ls(x) = {lim sup x}? and ?li(x) = {lim inf x+}? (generalizing the convergence of sequences on the Alexandrov cube and its dual). We consider the minimal topology Olsi extending the (unique) sequential topologies O?s (left) and O?li (right) generated by the convergences ?ls and ?li and establish a general hierarchy between all these topologies and the corresponding a priori and a posteriori convergences. In addition, we observe some special classes of algebras and, in particular, show that in (?,2)-distributive algebras we have limOlsi = lim?s = ?s, while the equality Olsi = ?s holds in all Maharam algebras. On the other hand, in some collapsing algebras we have a maximal (possible) diversity.

Locally Flat Vector Lattices

1980 ◽  
Vol 32 (4) ◽  
pp. 924-936 ◽  
Author(s):  
Marlow Anderson
Keyword(s):  
Boolean Algebra ◽  
Essential Extension ◽  
Boolean Algebras ◽  
Complete Boolean Algebra ◽  
Lattice Ordered Group ◽  
Ordered Group ◽  
Vector Lattices ◽  
Trivial Intersection ◽  
Image Position

Let G be a lattice-ordered group (l-group). If X ⊆ G, then letThen X’ is a convex l-subgroup of G called a polar. The set P(G) of all polars of G is a complete Boolean algebra with ‘ as complementation and set-theoretic intersection as meet. An l-subgroup H of G is large in G (G is an essential extension of H) if each non-zero convex l-subgroup of G has non-trivial intersection with H. If these l-groups are archimedean, it is enough to require that each non-zero polar of G meets H. This implies that the Boolean algebras of polars of G and H are isomorphic. If K is a cardinal summand of G, then K is a polar, and we write G = K⊞K'.

scholarly journals The strong closure of Boolean algebras of projections in Banach spaces

2004 ◽  
Vol 77 (3) ◽  
pp. 365-370 ◽  
Author(s):  
J. Diestel ◽  
W. J. Ricker
Keyword(s):  
Banach Space ◽  
Boolean Algebra ◽  
Banach Spaces ◽  
Sequence Space ◽  
Boolean Algebras ◽  
Complete Boolean Algebra ◽  
Special Cases

AbstractThis note improves two previous results of the second author. They turn out to be special cases of our main theorem which states: A Banach space X has the property that the strong closure of every abstractly σ-complete Boolean algebra of projections in X is Bade complete if and only if X does not contain a copy of the sequence space ℓ∞.

Weak Distributivity, A Problem of Von Neumann and the Mystery of Measurability

2006 ◽  
Vol 12 (2) ◽  
pp. 241-266 ◽  
Author(s):  
Bohuslav Balcar ◽  
Thomas Jech
Keyword(s):  
Boolean Algebra ◽  
Representation Theorem ◽  
Chain Condition ◽  
Boolean Algebras ◽  
Partial Ordering ◽  
Additive Measure ◽  
Boolean Operations ◽  
Von Neumann ◽  
Countable Chain Condition ◽  
Countable Chain

This article investigates the weak distributivity of Boolean σ-algebras satisfying the countable chain condition. It addresses primarily the question when such algebras carry a σ-additive measure. We use as a starting point the problem of John von Neumann stated in 1937 in the Scottish Book. He asked if the countable chain condition and weak distributivity are sufficient for the existence of such a measure.Subsequent research has shown that the problem has two aspects: one set theoretic and one combinatorial. Recent results provide a complete solution of both the set theoretic and the combinatorial problems. We shall survey the history of von Neumann's Problem and outline the solution of the set theoretic problem. The technique that we describe owes much to the early work of Dorothy Maharam to whom we dedicate this article.§1. Complete Boolean algebras and weak distributivity. ABoolean algebrais a setBwith Boolean operationsa˅b(join),a˄b(meet) and −a(complement), partial orderinga≤bdefined bya˄b=aand the smallest and greatest element,0and1. By Stone's Representation Theorem, every Boolean algebra is isomorphic to an algebra of subsets of some nonempty setS, under operationsa∪b,a∩b,S−a, ordered by inclusion, with0= ∅ and1=S.Complete Boolean algebras and weak distributivity.A Boolean algebrais a setBwith Boolean operationsa˅b(join),a˄b(meet) and -a(complement), partial orderinga≤bdefined bya˄b=aand the smallest and greatest element.0and1. By Stone's Representation Theorem, every Boolean algebra is isomorphic to an algebra of subsets of some nonempty setS, under operationsa∪b,a∩b,S-a, ordered by inclusion, with0= ϕ and1=S.

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