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Semi-Cohen Boolean algebras

Bohuslav Balcar, Thomas Jech, Jindřich Zapletal
1997 Annals of Pure and Applied Logic  
We investigate classes of Boolean algebras related to the notion of forcing that adds Cohen reals. A Cohen algebra is a Boolean algebra that is dense in the completion of a free Boolean algebra.  ...  We introduce and study generalizations of Cohen algebras: semi-Cohen algebras, pseudo-Cohen algebras and potentially Cohen algebras. These classes of Boolean algebras are closed under completion.  ...  For example, the forcing from [6] is similar to the one we just described.  ... 
doi:10.1016/s0168-0072(97)00009-2 fatcat:rwbtfxv53rb2tnrg6rwwsisvrm

Semi-Cohen Boolean algebras [article]

Bohuslav Balcar, Thomas Jech, Jindřich Zapletal
1995 arXiv   pre-print
We investigate classes of Boolean algebras related to the notion of forcing that adds Cohen reals. A >>Cohen algebra<< is a Boolean algebra that is dense in the completion of a free Boolean algebra.  ...  We introduce and study generalizations of Cohen algebras: semi-Cohen algebras, pseudo-Cohen algebras and potentially Cohen algebras. These classes of Boolean algebras are closed under completion.  ...  C κ is the completion of the free Boolean algebra on κ generators; equivalently, C κ is the algebra of all regular open subsets of the topological product space {0, 1} κ .  ... 
arXiv:math/9506208v1 fatcat:jncwdvuraffqlluzusdsvfo5gm

On absolutely divergent series [article]

Sakae Fuchino, Heike Mildenberger, Saharon Shelah, Peter Vojtas
1999 arXiv   pre-print
We show that in the aleph_2-stage countable support iteration of Mathias forcing over a model of CH the complete Boolean algebra generated by absolutely divergent series under eventual dominance is not  ...  isomorphic to the completion of P(omega)/fin.  ...  Acknowledgement: The authors would like to thank Andreas Blass very much for carefully reading a preliminary version of this paper, pointing out a gap, and making valuable suggestions.  ... 
arXiv:math/9903114v1 fatcat:yoqmqrhfancprbxgwlsdxhippe

Measurable cardinals and the cardinality of Lindelöf spaces [article]

Marion Scheepers
2010 arXiv   pre-print
If it is consistent that there is a measurable cardinal, then it is consistent that all points g-delta Rothberger spaces have "small" cardinality.  ...  A subset D of the Boolean algebra P(κ)/J is said to be dense if there is for each b ∈ P(κ)/J a d ∈ D with d < b.  ...  In Section 4 we describe models of set theory in which the Continuum Hypothesis fails while there is a "small" upper bound on the cardinality of points G δ indestructibly Lindelöf spaces.  ... 
arXiv:0909.3663v2 fatcat:evhiqog56vfafa5d5mumbuinm4

Measurable cardinals and the cardinality of Lindelöf spaces

Marion Scheepers
2010 Topology and its Applications  
We obtain from the consistency of the existence of a measurable cardinal the consistency of "small" upper bounds on the cardinality of a large class of Lindelöf spaces whose singletons are G δ sets.  ...  Tall, investigating Arhangel'skii's problem, defined the class of indestructibly Lindelöf spaces.  ...  Tall for very a stimulating correspondence regarding the problems treated in this paper, and for permission to include his argument (slightly adapted) in the proof of Theorem 13.  ... 
doi:10.1016/j.topol.2010.03.004 fatcat:duwnjyj2avbfzjaje67dlpvqei

Query languages with arithmetic and constraint databases

Leonid Libkin
1999 ACM SIGACT News  
An important class of collapse result is generic collapse, describing expressiveness with respect to`pure', or generic, queries.  ...  Families of sets arising in such a way are called de When the collapse fails We have seen one example of the failure of the natural-active collapse: N = hN; +; i.  ... 
doi:10.1145/337885.337894 fatcat:vpvwkahpkbhyhbxk5loodrdrme

Large Cardinals with Forcing [chapter]

Akihiro Kanamori
2012 Handbook of the History of Logic  
A more familiar conceptualization in mathematics, Tarski investigated the general notion of ideal on a Boolean algebra in place of the power set algebra P (Z).  ...  By establishing in ZFC that e.g. there is a complete Boolean algebra assigning the formula expressing ¬CH Boolean value one, a semantic construction was replaced by a syntactic one that directly secured  ... 
doi:10.1016/b978-0-444-51621-3.50004-9 fatcat:rxofgx3z6nelvlyhmdybzlox6e

Complexity of mixed equilibria in Boolean games [article]

Egor Ianovski
2017 arXiv   pre-print
To address this, the present work focuses on the complexity of algorithmic problems dealing with mixed strategies in Boolean games.  ...  Boolean games are a succinct representation of strategic games wherein a player seeks to satisfy a formula of propositional logic by selecting a truth assignment to a set of propositional variables under  ...  A key result of the authors is that Boolean games form a Boolean algebra modulo strategic equivalence, under the operations +, · and .  ... 
arXiv:1702.03620v1 fatcat:2gtsuorinbdnnnyupsh2jcuzfq

Polynomial clone reducibility

Quinn Culver
2013 Archive for Mathematical Logic  
A polynomial clone is a set of functions over a finite set X that is closed under composition and contains all the constant and projection functions.  ...  Polynomial-clone reducibilities are generalizations of the truth-table reducibilities.  ...  Using the so called truth-table cylinders, we show that there are times when all the continuum-many reducibilities collapse.  ... 
doi:10.1007/s00153-013-0351-x fatcat:yjlq4ooxhnd4fdu324pcndufuu

Mycielski ideals generated by uncountable systems

A. Rosłanowski
1993 Colloquium Mathematicum  
My thanks are due to Janusz Pawlikowski for his help in the preparation of this paper.  ...  For every countable system K the completion of the Boolean algebra BOREL(X ω )/M X ,K is isomorphic to the collapsing algebra Col(ω, c). P r o o f.  ...  Notions of forcing connected with C X and P X . In 2.5 we showed that for countable K the Boolean algebra BOREL(X ω )/M X ,K as a notion of forcing is equivalent to the collapsing algebra Col(ω, c).  ... 
doi:10.4064/cm-66-2-187-200 fatcat:z6it5sl6krajlf4vw2m3cqfx6u

Fine hierarchies and m-reducibilities in theoretical computer science

Victor L. Selivanov
2008 Theoretical Computer Science  
This is a survey of results about versions of fine hierarchies and many-one reducibilities that appear in different parts of theoretical computer science.  ...  These notions and related techniques play a crucial role in understanding complexity of finite and infinite computations.  ...  Of course, the DH over L may collapse (e.g., if L is a Boolean algebra then L = BC (L) and the DH over L collapses to the first level).  ... 
doi:10.1016/j.tcs.2008.06.031 fatcat:canuexaafjau3i4mcdpdu4oibe

Generic compactness reformulated

Bernhard K�nig
2004 Archive for Mathematical Logic  
One principle of pure reflection is introduced that is as strong as generic supercompactness of ω2 by σ-closed forcing.  ...  This new concept implies CH and extends the reflection principles for stationary sets in a canonical way.  ...  The article is the second part of my thesis [10] which was produced under supervision of Hans-Dieter Donder and Stevo Todorčević.  ... 
doi:10.1007/s00153-003-0211-1 fatcat:s4ck2fsntje2fkpp4tmqdiq7ly

Atom-canonicity, relativized representations and omitting types for clique guarded semantics and guarded logics [article]

Tarek Sayed Ahmed
2013 arXiv   pre-print
We study atom canonicity for several varieties of cylindric like algebras that contain properly the variety of representable algebras.  ...  In the second part of the paper various notions of representability originally formulated for atom structure are lifted in an obvious way to the algebra level, like weak and strong representability.  ...  One can alternatively show that A ≡ ∞ω B using "soft model theory" as follows: Form a Boolean extension M * of the universe M in which the cardinalities of A and B collapse to ω.  ... 
arXiv:1308.6165v1 fatcat:jlf6cbjycjbmfg7tqmjjgxu6gm

On completions, neat embeddings and omittings types, yet again [article]

Tarek Sayed Ahmed
2013 arXiv   pre-print
In this paper we investigate using the methodology of algebraic logic, deep algebraic results to prove three new omitting types theorems for finite variable fragments of first order logic.  ...  As a sample we show that, assuming the existence of certain finite relation algebras, that for any k∈ω, there exists ∈ RPEA_n∩_n_n+k such that Rd_ Sc∉ S_n_n+k+1.  ...  Letx enumerate nodes(N s ) ∩ nodes(N t ) Ifx is short, then there are at most two nodes in the intersection and this case is similar to the cylindrifier move, she uses ρ s for the suffixes of the red.  ... 
arXiv:1307.1016v1 fatcat:wvhibz354fedjolfyikgcqdci4

Quantum Information: The New Frontier [chapter]

Karl Svozil
2001 Unconventional Models of Computation, UMC'2K  
A brief, rather subjective outline is presented.  ...  Quantum information and computation is the new hype in physics. It is promising, mindboggling and even already applicable in cryptography, with good prospects ahead.  ...  Every partition E ∈ B can be identified with a Boolean algebra B E in a natural way by identifying the elements of the partition with the atoms of the Boolean algebra.  ... 
doi:10.1007/978-1-4471-0313-4_19 fatcat:3yr4aovxrrh5lkn5lgzudre2bm
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