The uniformly closed algebra generated by a complete Boolean algebra of projections

1989 ◽  
Vol 201 (3) ◽  
pp. 429-439 ◽  
Author(s):  
Werner J. Ricker ◽  
Helmut H. Schaefer
Keyword(s):  
Boolean Algebra ◽  
Complete Boolean Algebra ◽  
Closed Algebra ◽  
Uniformly Closed

Pseudocomplemented and Implicative Semilattices

1982 ◽  
Vol 34 (2) ◽  
pp. 423-437 ◽  
Author(s):  
C. S. Hoo
Keyword(s):  
Boolean Algebra ◽  
Closed Algebra ◽  
Implicative Semilattices

Let L be a semilattice and let a ∊ L. We refer the reader to Definitions 2.2, 2.4, 2.5 and 2.12 below for the terminology. If L is a-implicative, let Ca be the set of a-closed elements of L, and let Da be the filter of a-dense elements of L. Then Ca is a Boolean algebra. If a = 0, then C0 and D0 are the usual closed algebra and dense filter of L. If L is a-admissible and f : Ca × Da → Da is the corresponding admissible map, we can form a quotient semilattice Ca × D0f. In case a = 0, Murty and Rao [4] have shown that C0 × D0/f is isomorphic to L, and hence that C0 × D0 is 0-admissible. In case L is in fact implicative, Nemitz [5] has shown that C0 × D0/f is isomorphic to L, and that C0 × D0/f is also implicative.

Power-collapsing games

2008 ◽  
Vol 73 (4) ◽  
pp. 1433-1457 ◽  
Author(s):  
Miloš S. Kurilić ◽  
Boris Šobot
Keyword(s):  
Boolean Algebra ◽  
Dense Subset ◽  
Winning Strategy ◽  
Zero Element ◽  
The Other ◽  
Complete Boolean Algebra ◽  
Infinite Cardinal ◽  
Other Hand ◽  
Game Theoretic

AbstractThe game is played on a complete Boolean algebra , by two players. White and Black, in κ-many moves (where κ is an infinite cardinal). At the beginning White chooses a non-zero element p ∈ . In the α-th move White chooses pα ∈ (0, p) and Black responds choosing iα ∈{0, 1}. White winsthe play iff . where and .The corresponding game theoretic properties of c.B.a.'s are investigated. So, Black has a winning strategy (w.s.) if κ ≥ π() or if contains a κ-closed dense subset. On the other hand, if White has a w.s., then κ ∈ . The existence of w.s. is characterized in a combinatorial way and in terms of forcing. In particular, if 2<κ = κ ∈ Reg and forcing by preserves the regularity of κ, then White has a w.s. iff the power 2κ is collapsed to κ in some extension. It is shown that, under the GCH, for each set S ⊆ Reg there is a c.B.a. such that White (respectively. Black) has a w.s. for each infinite cardinal κ ∈ S (resp. κ ∉ S). Also it is shown consistent that for each κ ∈ Reg there is a c.B.a. on which the game is undetermined.

scholarly journals A Complete Boolean Algebra That Has No Proper Atomless Complete Subalgebra

1996 ◽  
Vol 182 (3) ◽  
pp. 748-755 ◽  
Author(s):  
Thomas Jech ◽  
Saharon Shelah
Keyword(s):  
Boolean Algebra ◽  
Complete Boolean Algebra

Complete Boolean ultraproducts

1987 ◽  
Vol 52 (2) ◽  
pp. 530-542
Author(s):  
R. Michael Canjar
Keyword(s):  
Boolean Algebra ◽  
Regular Cardinal ◽  
Complete Boolean Algebra ◽  
Full Structure ◽  
Truth Value ◽  
Image Position

Throughout this paper, B will always be a Boolean algebra and Γ an ultrafilter on B. We use + and Σ for the Boolean join operation and · and Π for the Boolean meet.κ is always a regular cardinal. C(κ) is the full structure of κ, the structure with universe κ and whose functions and relations consist of all unitary functions and relations on κ. κB is the collection of all B-valued names for elements of κ. We use symbols f, g, h for members of κB. Formally an element f ∈ κB is a mapping κ → B with the properties that Σα∈κf(α) = 1B and that f(α) · f(β) = 0B whenever α ≠ β. We view f(α) as the Boolean-truth value indicating the extent to which the name f is equal to α, and we will hereafter write ∥f = α∥ for f(α). For every α ∈ κ there is a canonical name fα ∈ κB which has the property that ∥fα = α∥ = 1. Hereafter we identify α and fα.If B is a κ+-complete Boolean algebra and Γ is an ultrafilter on B, then we may define the Boolean ultraproduct C(κ)B/Γ in the following manner. If ϕ(x0, x1, …, xn) is a formula of Lκ, the language for C(κ) (which has symbols for all finitary functions and relations on κ), and f0, f1, …, fn−1 are elements of κB then we define the Boolean-truth value of ϕ(f0, f1, …, fn−1) as

scholarly journals Realizing isomorphisms of category algebras

1978 ◽  
Vol 19 (1) ◽  
pp. 5-10 ◽  
Author(s):  
Dorothy Maharam ◽  
A.H. Stone
Keyword(s):  
Boolean Algebra ◽  
Metric Spaces ◽  
Analogous Result ◽  
Complete Boolean Algebra ◽  
Borel Sets ◽  
Complete Metric Spaces ◽  
First Category ◽  
Category Algebras

Let C(X) denote the complete boolean algebra of Borel sets modulo first category sets of the space X. Given an isomorphism τ between C(X) and C(Y), where X and Y are complete metric spaces, it is shown that there exists a homeomorphism T, between residual subsets A of X and B of Y, that induces τ. When X = Y one can make A = B. An analogous result is stated when τ is a complete isomorphism onto a subalgebra.

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.

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