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1. Boolean Algebra - Wikipedia, The Free Encyclopedia
Boolean algebra (logic), an equational theory of truth values. Boolean algebras canonically defined gives an alternative perspective on the structure
http://en.wikipedia.org/wiki/Boolean_algebra
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##### Boolean algebra
• Boolean , another disambiguation page.
This disambiguation page lists articles associated with the same title. If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " http://en.wikipedia.org/wiki/Boolean_algebra Categories Disambiguation Mathematical disambiguation Views Personal tools Navigation interaction Search Toolbox Languages

2. The Mathematics Of Boolean Algebra (Stanford Encyclopedia Of Philosophy)
Boolean algebra is the algebra of twovalued logic with only sentential connectives, or equivalently of algebras of sets under union and complementation.
http://plato.stanford.edu/entries/boolalg-math/
Cite this entry Search the SEP Advanced Search Tools ...
##### The Mathematics of Boolean Algebra
First published Fri 5 Jul, 2002
##### 1. Definition and simple properties
A Boolean algebra (BA) is a set A A such that the following laws hold: commutative and associative laws for addition and multiplication, distributive laws both for multiplication over addition and for addition over multiplication, and the following special laws:
These laws are better understood in terms of the basic example of a BA, consisting of a collection A of subsets of a set X closed under the operations of union, intersection, complementation with respect to X X x y if and only if x y y X to have just one element. An important elementary result is that an equation holds in all BAs if and only if it holds in the two-element BA. Next, we define x y x y y x ). Then

3. Robbins Algebras Are Boolean
A web text by William McCune describing the solution of this problem by a theoremproving program, with input files and the proofs.
http://www.cs.unm.edu/~mccune/papers/robbins/
##### Robbins Algebras Are Boolean
William McCune
Posted on the Web October, 1996. Last updated October, 2006. These Web pages contain some information on the solution of the Robbins problem. A paper on this topic appears in the Journal of Automated Reasoning [W. McCune, "Solution of the Robbins Problem", JAR 19(3), 263276 (1997)]. Here is a preprint . The JAR paper has simpler proofs than the ones below on this page. Here are the input files and proofs corresponding to the JAR paper
##### Introduction
The Robbins problem-are all Robbins algebras Boolean?-has been solved: Every Robbins algebra is Boolean. This theorem was proved automatically by EQP , a theorem proving program developed at Argonne National Laboratory.
##### Historical Background
In 1933, E. V. Huntington presented [1,2] the following basis for Boolean algebra: x + y = y + x. [commutativity] (x + y) + z = x + (y + z). [associativity] n(n(x) + y) + n(n(x) + n(y)) = x. [Huntington equation] Shortly thereafter, Herbert Robbins conjectured that the Huntington equation can be replaced with a simpler one : n(n(x + y) + n(x + n(y))) = x. [Robbins equation]

4. Boolean Algebra And Logic Circuits
Boolean Algebra and Logic Circuits. Nov24-2007. quick.gif. space2.gif. space2.gif. space2.gif. space2.gif space.gif. Boolean Switching algebras
http://www.asic-world.com/digital/boolean.html
 @import url(/css/main.css); @import url(/css/syntax.css); Boolean Algebra and Logic Circuits Dec-14-2007 Symbolic Logic Precedence Function Definitions Truth Tables ... Truth Table Web www.asic-world.com Do you have any Comment? mail me at: deepak@asic-world.com

5. Robbins Algebras Vs. Boolean Algebras
In the early 1930s, Huntington proposed several axiom systems for Boolean algebras. Robbins slightly changed one of them and asked if the resulted system is
http://www.cs.ualberta.ca/~piotr/Mizar/mirror/htdocs/JFM/Vol13/robbins1.html
Journal of Formalized Mathematics
Volume 13, 2001

University of Bialystok

Association of Mizar Users
##### Robbins Algebras vs. Boolean Algebras
University of Bialystok
##### Summary.
], Huntington [ ] and Dahn [
This work has been partially supported by TYPES grant IST-1999-29001.
##### MML Identifier:
The terminology and notation used in this paper have been introduced in the following articles [
##### Contents (PDF format)
• Preliminaries
• Pre-Ortholattices
• Correspondence between Boolean Pre-OrthoLattices and Boolean Lattices
• Proofs according to Bernd Ingo Dahn
• Proofs according to William McCune
##### Bibliography
1] Czeslaw Bylinski. Binary operations Journal of Formalized Mathematics
2] Czeslaw Bylinski. Functions and their basic properties Journal of Formalized Mathematics
3] Czeslaw Bylinski. Functions from a set to a set Journal of Formalized Mathematics
9] Michal Muzalewski. Midpoint algebras Journal of Formalized Mathematics
10] Michal Muzalewski. Construction of rings and left-, right-, and bi-modules over a ring Journal of Formalized Mathematics
11] Andrzej Trybulec. Tarski Grothendieck set theory Journal of Formalized Mathematics Axiomatics
12] Stanislaw Zukowski.
• 6. Boolean Algebra - Wolfram Demonstration Project
A Boolean algebra is the partial order on subsets defined by inclusion. Boolean algebras form lattices and have a recursive structure apparent in their
http://demonstrations.wolfram.com/BooleanAlgebra/
##### Boolean Algebra
loadFlash(494, 487, 'BooleanAlgebra'); A Boolean algebra is the partial order on subsets defined by inclusion. Boolean algebras form lattices and have a recursive structure apparent in their Hasse diagrams. The Hasse diagram for a Boolean algebra of order illustrates the partition between left and right halves of the lattice, each of which is the Boolean algebra on elements.
" Boolean Algebra " from The Wolfram Demonstrations Project http://demonstrations.wolfram.com/BooleanAlgebra/

7. Boolean.Algebra
This is the main module of the Boolean hierachy and provides a class which abstracts common operations on Boolean algebras. note, we redefine some prelude
http://repetae.net/recent/src/hsdocs/Boolean-Algebra.html
 Contents Index Boolean.Algebra Description This is the main module of the Boolean hierachy and provides a class which abstracts common operations on boolean algebras. note, we redefine some prelude functions, but the new definitons mean the same thing for Bool so it will not hurt existing code. to use properly: Synopsis class SemiBooleanAlgebra a where class SemiBooleanAlgebra BooleanAlgebra a where true :: a false :: a not xor and or ... BooleanAlgebra Documentation class SemiBooleanAlgebra a where This class is mainly for syntax re-use, there are many types which are very similar to boolean algebras, but do not have suitable distinguished values to choose for true and false. and should be strict only in their first argument, and return one of their arguments if possible. Methods Instances SemiBooleanAlgebra Bool SemiBooleanAlgebra Int SemiBooleanAlgebra Integer SemiBooleanAlgebra a, SemiBooleanAlgebra SemiBooleanAlgebra (a, b) SemiBooleanAlgebra SemiBooleanAlgebra SemiBooleanAlgebra Boolean a) SemiBooleanAlgebra Fuzz a) SemiBooleanAlgebra (Maybe a) SemiBooleanAlgebra [a] (Monad m

8. Handbook Of Boolean Algebras - Elsevier
This Handbook treats those parts of the theory of Boolean algebras of most interest to pure mathematicians the settheoretical abstract theory and
http://www.elsevier.com/wps/product/cws_home/501441
 Home Site map Elsevier websites Alerts ... Handbook of Boolean Algebras Book information Product description Author information and services Ordering information Bibliographic information Conditions of sale Volume information Volume 1 Volume 2 Book-related information Submit your book proposal Other books in same subject area About Elsevier Select your view HANDBOOK OF BOOLEAN ALGEBRAS http://books.elsevier.com/elsevier/?isbn=044470261X Edited by J.D. Monk , with the cooperation of R. Bonnet Description This Handbook treats those parts of the theory of Boolean algebras of most interest to pure mathematicians: the set-theoretical abstract theory and applications and relationships to measure theory, topology, and logic. It is divided into two parts (published in three volumes). Part I (volume 1) is a comprehensive, self-contained introduction to the set-theoretical aspects of the theory of Boolean Algebras. It includes, in addition to a systematic introduction of basic algebra and topological ideas, recent developments such as the Balcar-Franek and Shelah-Shapirovskii results on free subalgebras. Part II (volumes 2 and 3) contains articles on special topics describing - mostly with full proofs - the most recent results in special areas such as automorphism groups, Ketonen's theorem, recursive Boolean algebras, and measure algebras. Volumes Volume 1 Volume 2 Handbook of Boolean Algebras, Volume 2

9. Boolean Algebras And Natural Language: A Measurement Theoretic Approach
Boolean algebras and Natural Language A Measurement Theoretic Approach.
http://www.hf.uio.no/ifikk/filosofi/njpl/vol4no2/alglang/index.html
Next: I Up: Contents
Eli Dresner
##### Abstract:
In the first section of this paper I discuss several accounts of the way Boolean algebra applies to the logical sentential connectives of natural languages. In the second section I present a superior account, modeled after the use of numbers in measurement.
##### Footnotes
I would like to thank Charles Chihara, Donald Davidson, Stephen Neale, Chris Pincock and Bruce Vermazen for their comments. Also, I thank the referee of the Nordic Journal of Philosophical Logic , whose comments greatly improved this paper.

Postscript, compressed: alglang-ps.zip (62196 bytes), alglang.ps.gz (62073 bytes)
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Nordic Journal of Philosophical Logic, Vol. 4, No. 2, pp. 175189.

10. DSpace At MIT: Quantifier-Free Boolean Algebra With Presburger Arithmetic Is NP-
Abstract, Boolean Algebra with Presburger Arithmetic (BAPA) combines1) Boolean algebras of sets of uninterpreted elements (BA)and 2) Presburger arithmetic
http://hdl.handle.net/1721.1/35258
 About DSpace Software Search DSpace Advanced Search Home Browse Communities Titles Authors Subjects ... By Date Sign on to: Receive email updates My DSpace authorized users Edit Profile General Help About DSpace@MIT DSpace at MIT ... CSAIL Technical Reports (July 1, 2003 - present) Please use this identifier to cite or link to this item: http://hdl.handle.net/1721.1/35258 Title: Quantifier-Free Boolean Algebra with Presburger Arithmetic is NP-Complete Authors: Kuncak, Viktor Advisor: Martin Rinard Other contributors: Computer Architecture Keywords: Caratheodory theorem integer linear programming integer cone Hilbert basis Issue Date: 1-Jan-2007 Series/Report no.: Massachusetts Institute of Technology Computer Science and Artificial Intelligence Laboratory Abstract: Boolean Algebra with Presburger Arithmetic (BAPA) combines1) Boolean algebras of sets of uninterpreted elements (BA)and 2) Presburger arithmetic operations (PA). BAPA canexpress the relationship between integer variables andcardinalities of unbounded finite sets and can be used toexpress verification conditions in verification of datastructure consistency properties.In this report I consider the Quantifier-Free fragment ofBoolean Algebra with Presburger Arithmetic (QFBAPA).Previous algorithms for QFBAPA had non-deterministicexponential time complexity. In this report I show thatQFBAPA is in NP, and is therefore NP-complete. My resultyields an algorithm for checking satisfiability of QFBAPAformulas by converting them to polynomially sized formulasof quantifier-free Presburger arithmetic. I expect thisalgorithm to substantially extend the range of QFBAPAproblems whose satisfiability can be checked in practice.

 11. Relationships Between Post And Boolean Algebras With Application To Multi-Valued The fundamentals of Post algebras are presented along with extensions that will be useful in a proposed multivalued switching theory.http://stinet.dtic.mil/oai/oai?verb=getRecord&metadataPrefix=html&identifier=AD0

12. Finite Boolean Algebras And Subgroup Lattices Of Finite Abelian Groups
Arguments involving both complementation in a Boolean algebra as well as the distributivity of a lattice are used. Prerequisites include an elementary
http://www.jyi.org/volumes/volume3/issue1/articles/watkins.html
 Journal of Young Investigators Undergraduate, Peer-Reviewed Science Journal Volume Three RESEARCH ARTICLE RECENT ISSUES ARCHIVES RESOURCES JYI NEWS ... ABOUT JYI Issue 1, March 2001 Finite Boolean Algebras and Subgroup Lattices Of Finite Abelian Groups Ryan Watkins Abilene Christian University Advisor: Jason Holland, Ph.D. Assistant Professor of Mathematics Abilene Christian University Abstract We use the fundamental theorem of finite abelian groups to give a characterization of when the subgroup lattice of a finite abelian group can be given the structure of a Boolean algebra. Arguments involving both complementation in a Boolean algebra as well as the distributivity of a lattice are used. Prerequisites include an elementary course in abstract algebra as well as familiarity with elementary lattice theory. Introduction We shall follow the notation in and suggest this as a reference for unexplained terminology as well. We consider only finite abelian groups in this paper. We use the notation Z

13. Algebraic Partial Boolean Algebras
Partial Boolean algebras, first studied by Kochen and Specker in the 1960s, provide the structure for BellÂKochenÂSpecker theorems which deny the existence
http://www.iop.org/EJ/abstract/0305-4470/36/13/319
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##### Algebraic partial Boolean algebras
Derek Smith J. Phys. A: Math. Gen. 3899-3910 doi:10.1088/0305-4470/36/13/319 PDF (164 KB) References Articles citing this article
Derek Smith

Math Department, Lafayette College, Easton, PA 18042, USA
E-mail: smithder@lafayette.edu Abstract. The main result of this paper is that every algebraic finitely-generated partial Boolean algebra B T ) is finite when the underlying space is three-dimensional, answering a question of Kochen and showing that Conway and Kochen's infinite algebraic partial Boolean algebra has minimum dimension. This result contrasts the existence of an infinite (non-algebraic) B T ) generated by eight elements in an abstract orthomodular lattice of height 3. We then initiate a study of higher-dimensional algebraic partial Boolean algebras. First, we describe a restriction on the determinants of the elements of B T ) that are generated by a given set T . We then show that when the generating set T consists of the rays spanning the minimal vectors in a real irreducible root lattice

14. Category:Boolean Algebra - Indopedia, The Indological Knowledgebase
Boolean algebra is a simple mathematical model of elementary formal logic. Stone s representation theorem for Boolean algebras
http://www.indopedia.org/Category:Boolean_algebra.html
Categories
Algebra Abstract algebra Logic ... Wikipedia Article
##### Category:Boolean algebra
Ã Â¤ÂÃ Â¥ÂÃ Â¤ÂÃ Â¤Â¾Ã Â¤Â¨Ã Â¤ÂÃ Â¥ÂÃ Â¤Â¶: - The Indological Knowledgebase Boolean algebra is a simple mathematical model of elementary formal logic
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##### Articles in category "Boolean algebra"
There are 17 articles in this category.
##### T

15. Springer Online Reference Works
The applications of Boolean algebras to logic are based on the interpretation of the Boolean algebras are used in the foundations of probability theory.
http://eom.springer.de/b/b016920.htm
 Encyclopaedia of Mathematics B Article referred from Article refers to Boolean algebra, Boolean lattice A partially ordered set of a special type. It is a distributive lattice , which satisfies the relations The operations sup and inf are usually denoted by the symbols and , and sometimes by and respectively, in order to stress their similarity to the set-theoretical operations of union and intersection. The notation or may be employed instead of . The complement of an element in a Boolean algebra is unique. A Boolean algebra can also be defined in a different manner. Viz. as a non-empty set with the operations which satisfy the following axioms: If this approach is adopted, the order is not assumed to be given in advance, and is introduced by the following condition: if and only if In addition to the basic operations , other operations in a Boolean algebra can be defined; among these the symmetric difference operation is particularly important: Alternative notations are Any Boolean algebra is a Boolean ring ); any Boolean ring with a unit element can be considered as a Boolean algebra.

16. [math/9506208] Semi-Cohen Boolean Algebras
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
http://arxiv.org/abs/math/9506208
##### arXiv.orgmath
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##### Title: Semi-Cohen Boolean algebras
Authors: Bohuslav Balcar Thomas Jech (Submitted on 17 Jun 1995) Abstract: Subjects: Logic (math.LO) Report number: Logic E-prints June 17, 1995 Cite as: arXiv:math/9506208v1 [math.LO]
##### Submission history
From: Thomas Jech [ view email
Sat, 17 Jun 1995 00:00:00 GMT (21kb)
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17. Atlas: Boolean Algebra And Lambda Calculus By Antonino Salibra
One of the milestones of modern algebra is the Stone representation theorem for Boolean algebras. In 1 Manzonetto and Salibra have shown that Comer s
http://atlas-conferences.com/cgi-bin/abstract/caug-81
 Atlas home Conferences Abstracts about Atlas ALGEBRAIC AND TOPOLOGICAL METHODS IN NON-CLASSICAL LOGICS III (TANCL'07) August 5-9, 2007 St Anne's College, University of Oxford Oxford, England Organizers Mai Gehrke and Hilary Priestley View Abstracts Conference Homepage Boolean algebra and lambda calculus by Antonino Salibra University of Venice One of the milestones of modern algebra is the Stone representation theorem for Boolean algebras. In  Manzonetto and Salibra have shown that Comer's generalization of the Stone representation theorem holds also for combinatory algebras: any combinatory algebra is isomorphic to a weak Boolean product of directly indecomposable combinatory algebras. The proof of the representation theorem for combinatory algebras is based on the fact that every combinatory algebra has central elements, i.e., elements which define a direct decomposition of the algebra as the Cartesian product of two other combinatory algebras. Central elements in a combinatory algebra constitute a Boolean algebra, whose Boolean operations can be defined by suitable combinators. Then it is natural to investigate the semantics of lambda calculus given in terms of models, which are directly indecomposable as combinatory algebras (indecomposable semantics, for short). The indecomposable semantics includes all known models of lambda calculus (i.e., the continuous, stable and strongly stable semantics), and the term models of all semisensible lambda theories (theories which do not equate solvable and unsolvable terms). The class of indecomposible combinatory algebras is a universal class, so that it is closed under subalgebras and ultraproducts. This implies that, if the term model of a lambda theory T is decomposible in a non-trivial way, then every model of T can be also decomposible in a non-trivial way. It follows that the indecomposable semantics is incomplete because there exist lambda theories whose term models admit non-trivial central elements.

18. Boolean Algebra, Boolean Algebras- WordWeb Dictionary Definition
Derived forms Boolean algebras. Type of formal logic, mathematical logic, symbolic logic. Encyclopedia Boolean algebra. Nearest
http://www.wordwebonline.com/en/BOOLEANALGEBRA
##### WordWeb Online
Dictionary and Thesaurus: WordWeb Help Us Improve Links Look up words in any Windows program:
Noun: Boolean algebra
• A system of symbolic logic devised by George Boole ; used in computers
Boolean logic
Derived forms: Boolean algebras Type of: formal logic mathematical logic symbolic logic Encyclopedia: Boolean algebra Nearest bookplate
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Boolean algebra Boolean expression Boolean logic boolean operation Boolean search ... boomerang
• 19. Boolean Algebras - Hutchinson Encyclopedia Article About Boolean Algebras
Hutchinson encyclopedia article about Boolean algebras. Boolean algebras. Information about Boolean algebras in the Hutchinson encyclopedia.
http://encyclopedia.farlex.com/Boolean algebras
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##### Boolean algebras
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20. Book Boolean Algebras In Analysis (mathematics Its Applications, Vol. 540), Unde
book undergraduate level (part ii) engineering colleges et postgraduate - research - viva foreword to the english translation. denis artem evich
http://www.lavoisier.fr/notice/gb097645.html
 Search on All Book CD-Rom eBook Software The french leading professional bookseller Description Approximate price Author(s) : VLADIMIROV D.A Publication date : 04-2002 Language : ENGLISH Status : In Print (Delivery time : 10 days) Contents Foreword to the English Translation. Denis Artem'evich Vladimirov (1929-1994). Preface. Introduction. Part I: General Theory of Boolean Algebras. 0. Preliminaries on Boolean Algebras. 1. The Basic Apparatus. 2. Complete Boolean Algebras. 3. Representation of Boolean Algebras. 4. Topologies on Boolean Algebras. 5. Homomorphisms. 6. Vector Lattices and Boolean Algebras. Part II: Metric Theory of Boolean Algebras. 7. Normed Boolean Algebras. 8. Existence of a Measure. 9. Structure of a Normed Boolean Algebra. 10. Independence. Appendices. Prerequisites to Set Theory and General Topology. 1. General remarks. 2. Partially ordered sets. 3. Topologies. Basics of Boolean Valued Analysis. 1. General remarks. 2. Boolean valued models. 3. Principles of Boolean valued analysis. 4. Ascending and descending. References. Index. Description Summary Foreword to the English Translation. Denis Artem'evich Vladimirov (1929-1994). Preface. Introduction. Part I: General Theory of Boolean Algebras. 0. Preliminaries on Boolean Algebras. 1. The Basic Apparatus. 2. Complete Boolean Algebras. 3. Representation of Boolean Algebras. 4. Topologies on Boolean Algebras. 5. Homomorphisms. 6. Vector Lattices and Boolean Algebras. Part II: Metric Theory of Boolean Algebras. 7. Normed Boolean Algebras. 8. Existence of a Measure. 9. Structure of a Normed Boolean Algebra. 10. Independence. Appendices. Prerequisites to Set Theory and General Topology. 1. General remarks. 2. Partially ordered sets. 3. Topologies. Basics of Boolean Valued Analysis. 1. General remarks. 2. Boolean valued models. 3. Principles of Boolean valued analysis. 4. Ascending and descending. References. Index.

 21. JSTOR Countable Boolean Algebras And Decidability. Countable Boolean algebras and decidability. English translation of Schetnye bulevy algebry i razreshimost . Siberian school of algebra and logic.http://links.jstor.org/sici?sici=0022-4812(199809)63:3<1188:CBAAD>2.0.CO;2-G

 22. Boolean Rings And Boolean Algebras Boolean rings and Boolean algebras Set Theory, Logic, Probability, Statistics.http://www.physicsforums.com/showthread.php?t=201036

23. CiteULike: α -cut-complete Boolean Algebras
Let A be a Boolean algebra, and \$ \$ an infinite cardinal number or the symbol \$ \$. An \$ \$cut in A is an ordered pair (F,H) of subsets of A,
http://www.citeulike.org/user/thstoeber/article/1421837
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Algebra Universalis , Vol. 39, No. 1. (2 July 1998), pp. 57-70. Citation format: Plain APA Chicago Elsevier Harvard MLA Nature Oxford Science Turabian Vancouver
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All tags in thstoeber's library Filter: test CiteULike organises scholarly (or academic) papers or literature and provides bibliographic (which means it makes bibliographies) for universities and higher education establishments. It helps undergraduates and postgraduates. People studying for PhDs or in postdoctoral (postdoc) positions. The service is similar in scope to EndNote or RefWorks or any other reference manager like BibTeX, but it is a social bookmarking service for scientists and humanities researchers.

 24. A Mathematician’s Scratchpad Â» Blog Archive Â» Boolean To C* Algebras The categories of Boolean algebras, Compact Hausdorff spaces and C* algebras respectively. S is the Stone space functor, C is the Âcontinuous functions onhttp://david.efnet-math.org/?p=10

25. Degree Spectra Of Relations On Boolean Algebras
We show that every computable relation on a computable Boolean algebra B is either definable by a quantifierfree formula with constants from B (in which
http://math.uchicago.edu/~drh/boolean.html
##### by Rod G. DowneySergey S. Goncharov , and Denis R. Hirschfeldt
Status: published in Algebra and Logic , vol. 42 (2003), pp. 105 - 111.
Availability: DVI PostScript , and PDF
Abstract. We show that every computable relation on a computable Boolean algebra B is either definable by a quantifier-free formula with constants from B (in which case it is obviously intrinsically computable) or has infinite degree spectrum.
drh@math.uchicago.edu

26. PlanetMath: Generalized Boolean Algebra
Clearly, a Boolean algebra is a generalized Boolean algebra. 06E99 (Order, lattices, ordered algebraic structures Boolean algebras Miscellaneous)
http://planetmath.org/encyclopedia/GeneralizedBooleanAlgebra.html
 (more info) Math for the people, by the people. Encyclopedia Requests Forums Docs ... RSS Login create new user name: pass: forget your password? Main Menu sections EncyclopÂ¦dia Papers Books Expositions meta Requests Orphanage Unclass'd Unproven ... Classification talkback Polls Forums Feedback Bug Reports downloads Snapshots PM Book information News Docs Wiki ChangeLog ... About generalized Boolean algebra (Definition) A lattice is called a generalized Boolean algebra if is distributive is relatively complemented , and has as the bottom Clearly, a Boolean algebra is a generalized Boolean algebra. Conversely, a generalized Boolean algebra with a top is a Boolean algebra, since is a bounded distributive complemented lattice , so each element has a unique complement by distributivity . So is a unary operator on which makes into a de Morgan algebra . A complemented de Morgan algebra is, as a result, a Boolean algebra. As an example of a generalized Boolean algebra that is not Boolean , let be an infinite set and let be the set of all finite subsets of . Then is generalized Boolean: order by inclusion , then is a distributive as the operation is inherited from , the powerset of . It is also relatively complemented: if where , then is the relative complement of in . Finally

27. IngentaConnect On Poset Boolean Algebras
The poset Boolean algebra of P, denoted F(P), is defined as follows The set Keywords poset algebras; superatomic Boolean algebras; scattered posets;
http://www.ingentaconnect.com/content/klu/orde/2003/00000020/00000003/05139036
var tcdacmd="dt";

28. On Some Small Cardinals For Boolean Algebras
(3) There is an atomless Boolean algebra A such that (A)= and (A)= . (4) If is also regular, then there is an atomless Boolean algebra A such that
http://projecteuclid.org/handle/euclid.jsl/1096901761
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##### On some small cardinals for Boolean algebras
Ralph McKenzie and J. Donald Monk Source: J. Symbolic Logic Volume 69, Issue 3 (2004), 674-682.
##### References
B. Balcar and P. Simon Disjoint refinement Handbook of Boolean algebras. Vol. 2

 29. Rough Operations On Boolean Algebras In this paper, we introduce two pairs of rough operations on Boolean algebras. First we define a pair of rough approximations based on a partition of thehttp://portal.acm.org/citation.cfm?id=1090346.1090351&coll=GUIDE&dl=GUIDE&CFID=9

30. Boolean Algebra
Stone Representation Theorem for Boolean algebras Each Boolean algebra is represented as the algebra of clopen sets of its spectrum, a coherent Hausdorff
http://orion.math.iastate.edu/jdhsmith/class/M567Defn.htm
##### Key concepts for Boolean Algebra
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Semigroup ( S
Set S with associative multiplication: x y z x y z
Semilattice ( S
Commutative ( x y y x ) and idempotent ( x x x ) semigroup.
Meet semilattice ( S
Semilattice with order x y iff x x y
Join semilattice ( S
Semilattice with order x y iff y x y
Lattice ( S
Set S with join semilattice structure ( S , + ) and meet semilattice structure ( S x y iff x y
Absorptive laws
x x y x and x x y x
Monoid ( S e
Semigroup ( S e x x x e
Bounded semilattice ( S e
Commutative idempotent monoid.
Bounded meet semilattice ( S
x 1 for all x in S
Bounded join semilattice ( S
x for all x in S
Bounded lattice ( S
x 1 for all x in S
Distributive lattice ( S
x y z x y x z ) and x y z x y x z
Complement (in bounded distributive lattice)
x x = and x x
Boolean algebra ( S
Bounded distributive lattice ( S x x x x = and x x
De Morgan laws
x y x y ) ' and ( x y x y
Symmetric difference/exclusive or (in Boolean algebra)
x xor y x y x y
Boolean ring
Ring with 1 and x x
Implication in Boolean algebra
c a c a
Heyting algebra ( S
Bounded lattice with ( c x a iff x c a
Identities for a Heyting algebra ( S
Bounded lattice identities, together with:

31. Boolean Algebras
2.3.1 Boolean algebras. Source Perkal. The power set of a set X. Let X b a set and let (X) be the collection of all subsets of X. If X = {x, y},
http://www.stenmorten.com/English/dm/231_bool.htm
2.3 Discrete mathematics
##### 2.3.1 Boolean algebras
Source: Perkal The power set of a set X. Let X b a set and let Perkal , x , ..., x n , and then include or exclude x , and so on. Thus there are 2 choices associated with x , and then 2 choices associated with x X) is called a power set of X.)" Operations on X)
Binary
- union
- intersection Unary
' - complementation Special elements
- the empty set
X - the whole or universal set X);
##### Truth tables
(If you know Norwegian, you might want to compare the OR (+) - table to this table in " Logikk og argumentasjon ".) x y x + y (or) x * y (and) This definition of + and * may seem a bit artificial. Note that + is the same as OR, and also the same as UNION, and that * is the same as AND, and the same as INTERSECTION. Thinking of + and * in these terms may be a bit more intuitive. If you think of as 'false', and 1 as 'true', is "0 AND 1" true? No, something in there is false, so "0 AND 1" is false. But is "0 OR 1" true? Yes, "0 OR 1" is true. In the Standard True/False Model, the ' is often replaced by a bar over the variable. Hence, when x =

32. Boolean Algebra -- From Wolfram MathWorld
A Boolean algebra is a mathematical structure that is similar to a Boolean ring, but that is defined using the meet and join operators instead of the usual
http://mathworld.wolfram.com/BooleanAlgebra.html
 Search Site Algebra Applied Mathematics Calculus and Analysis ... Boolean Algebras Boolean Algebra A Boolean algebra is a mathematical structure that is similar to a Boolean ring , but that is defined using the meet and join operators instead of the usual addition and multiplication operators. Explicitly, a Boolean algebra is the partial order on subsets defined by inclusion (Skiena 1990, p. 207), i.e., the Boolean algebra of a set is the set of subsets of that can be obtained by means of a finite number of the set operations union OR intersection AND ), and complementation NOT ) (Comtet 1974, p. 185). A Boolean algebra also forms a lattice (Skiena 1990, p. 170), and each of the elements of is called a Boolean function . There are Boolean functions in a Boolean algebra of order (Comtet 1974, p. 186). In 1938, Shannon proved that a two-valued Boolean algebra (whose members are most commonly denoted and 1, or false and true) can describe the operation of two-valued electrical switching circuits. In modern times, Boolean algebra and Boolean functions are therefore indispensable in the design of computer chips and integrated circuits.

33. Boolean Algebra --Â  Britannica Online Encyclopedia
Britannica online encyclopedia article on Boolean algebra symbolic system of mathematical logic that represents relationships between entitieseither ideas
http://www.britannica.com/eb/article-9080665/Boolean-algebra
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##### Boolean algebra
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34. Boolean Algebra
then we have the equations of Boolean algebra. Before 1900 Boolean algebra really meant the juggling of equations (and negequations) to reflect valid
http://www.math.uwaterloo.ca/~snburris/htdocs/scav/boolean/boolean.html
Previous: Comparing the expressive power ... Next: Second proof of compactness for propositional logic Up: Supplementary Text Topics
##### Boolean algebra
If we take the equations that are true in the the calculus of classes and replace the symbols using the following table
then we have the equations of Boolean algebra . Before 1900 Boolean algebra really meant the juggling of equations (and neg-equations) to reflect valid arguments. In 1904 E.V. Huntington wrote a paper  in which he viewed Boolean algebras as algebraic structures satisfying the equations obtained from the calculus of classes. This viewpoint became dominant in the 1920's under the influence of M.H. Stone and A. Tarski. Stone was initially interested in Boolean algebras in order to gain insight into the structure of rings of functions which were being investigated in functional analysis. He wrote two massive papers, one on the equivalence of Boolean algebras and Boolean rings, and the other on the duality between Boolean algebras and Boolean spaces [= totally disconnected compact Hausdorff spaces]. Tarski studied Boolean algebras while working on the abstract notion of `closure under deductive consequence'. In the 1930's Stone proved that every Boolean algebra is isomorphic to a field of sets, and that the equations true of the two-element Boolean algebra are the same as the equations true of all Boolean algebras; and these equations were consequences of a small initial set of defining equations. What has the modern subject of Boolean algebra got to do with propositional logic? Not very much. Boolean algebra became a deep and fascinating subject in its own right, with much more to offer than a convenient notation to analyze simple chains of reasoning. Nonetheless on the level of equivalence and equations the subjects of propositional logic, calculus of classes, and Boolean algebras are essentially the same, as illustrated by the following table:

35. Boolean Algebra
One tool to reduce logical expressions is the mathematics of logical expressions, introduced by George Boole in 1854 and known today as Boolean Algebra.
http://www.play-hookey.com/digital/boolean_algebra.html
 Home www.play-hookey.com Mon, 12-24-2007 Digital Logic Families Digital Experiments Analog ... Test HTML Direct Links to Other Digital Pages: Combinational Logic: Basic Gates Derived Gates The XOR Function Binary Addition ... Boolean Algebra Sequential Logic: RS NAND Latch RS NOR Latch Clocked RS Latch RS Flip-Flop ... Converting Flip-Flop Inputs Alternate Flip-Flop Circuits: D Flip-Flop Using NOR Latches CMOS Flip-Flop Construction Counters: Basic 4-Bit Counter Synchronous Binary Counter Synchronous Decimal Counter Frequency Dividers ... The Johnson Counter Registers: Shift Register (S to P) Shift Register (P to S) The 555 Timer: 555 Internals and Basic Operation 555 Application: Pulse Sequencer Boolean Algebra One of the primary requirements when dealing with digital circuits is to find ways to make them as simple as possible. This constantly requires that complex logical expressions be reduced to simpler expressions that nevertheless produce the same results under all possible conditions. The simpler expression can then be implemented with a smaller, simpler circuit, which in turn saves the price of the unnecessary gates, reduces the number of gates needed, and reduces the power and the amount of space required by those gates. One tool to reduce logical expressions is the mathematics of logical expressions, introduced by George Boole in 1854 and known today as

36. BOOLEAN ALGEBRA : Volume IV - Digital
An introduction to Boolean algebra from the perspective of electronic engineering.
Hilite.elementid = "main"; All About Circuits Search this site
##### BOOLEAN ALGEBRA

37. Elements Of Boolean Algebra
Laws of Boolean Algebra Laws of Boolean Algebra. Commutative Law; Associative Law; Distributive Law; Identity Law; Redundance Law; De Morgan s Theorem
http://www.ee.surrey.ac.uk/Projects/Labview/boolalgebra/index.html
##### Boolean Algebra
Introduction Laws of Boolean Algebra
• Commutative Law
• Associative Law ... On-line Quiz
##### Introduction
The most obvious way to simplify Boolean expressions is to manipulate them in the same way as normal algebraic expressions are manipulated. With regards to logic relations in digital forms, a set of rules for symbolic manipulation is needed in order to solve for the unknowns.
A set of rules formulated by the English mathematician George Boole describe certain propositions whose outcome would be either true or false . With regard to digital logic, these rules are used to describe circuits whose state can be either, 1 (true) or (false) . In order to fully understand this, the relation between the AND gate OR gate and NOT gate operations should be appreciated. A number of rules can be derived from these relations as Table 1 demonstrates.
• P1: X = or X = 1
Table 1: Boolean Postulates
##### Laws of Boolean Algebra
Table 2 shows the basic Boolean laws. Note that every law has two expressions, (a) and (b). This is known as duality . These are obtained by changing every AND(.) to OR(+), every OR(+) to AND(.) and all 1's to 0's and vice-versa.

 38. Boolean Algebra From FOLDOC Strangely, a Boolean algebra (in the mathematical sense) is not strictly an algebra, but is in fact a lattice. A Boolean algebra is sometimes defined as ahttp://foldoc.org/?Boolean algebra

39. Boolean Algebra - Definition Of Boolean Algebra By The Free Online Dictionary, T
Definition of Boolean algebra in the Online Dictionary. Meaning of Boolean algebra. What does Boolean algebra mean? Boolean algebra synonyms, Boolean
http://www.thefreedictionary.com/Boolean algebra
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##### Boolean algebra
Also found in: Encyclopedia Wikipedia Hutchinson 0.09 sec. Boolean algebra n. An algebra in which elements have one of two values and the algebraic operations defined on the set are logical OR, a type of addition, and logical AND, a type of multiplication. Boolean algebra (b l n) A form of symbolic logic, in which variables, which stand for propositions, have only the values "true" (or "1") and "false" (or "0"). Relationships between these values are expressed by the Boolean operators AND, OR, and NOT. For example, "a + b" means "a OR b", and its value is true as long as either a is true or b is true (or both). Boolean logic can be used to solve logical problems, and provides the mathematical tools fundamental to the design of digital computers. It is named after the mathematician George Boole. Also called Boolean logic . See also logic gate Thesaurus Legend: Synonyms Related Words Antonyms Noun Boolean algebra - a system of symbolic logic devised by George Boole; used in computers

40. Electronics/Boolean Algebra - Wikibooks, Collection Of Open-content Textbooks
Boolean Algebra was created by George Boole (1815 1864) in his paper An Investigation of the Laws of Thought, on Which Are Founded the Mathematical
http://en.wikibooks.org/wiki/Electronics/Boolean_Algebra
var wgNotice = ""; var wgNoticeLocal = ""; var wgNoticeLang = "en"; var wgNoticeProject = "wikibooks";
##### Contents
• Boolean Algebra
• Formal Mathematical Operators Boolean Algebra Laws
##### edit Boolean Algebra
Boolean Algebra was created by George Boole (1815 - 1864) in his paper An Investigation of the Laws of Thought, on Which Are Founded the Mathematical Theories of Logic and Probabilities , published in 1854. It had few applications at the time, but eventually scientists and engineers realized that his system could be used to create efficient computer logic. The Boolean system has two states: True (T) or False (F). This can be represented in several different ways as on or off, one or zero, yes or no, etc. These states are manipulated by three fundamental operations called logical operators AND OR and NOT . These operators take certain inputs and produce an output based on a predetermined table of results. For example, the AND operator takes two (or more) inputs and returns an 'on' result only when both (or all) inputs are 'on'.
• In these tables T means "True", or "Yes", or 1 (in electronics), and

41. Boolean Algebra, Mathematical Logic, Math, Online Tutoring - Tutorvista.com
An English mathematician, named George Boole invented this new kind of algebra which analyses logic mathematically. This Boolean algebra provided a basic
http://www.tutorvista.com/content/math/boolean-algebra/boolean-algebra.php
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##### Boolean Algebra
Mathematical Logic The study of logic through the use of mathematical symbols is called Mathematical Logic. Mathematical logic is also known as Symbolic Logic or Boolean Logic. Venn diagrams are used very frequently on problems of "set theory". Venn diagrams can also be used for deciding the truthfulness of statements. Boolean Algebra An English mathematician, named George Boole invented this new kind of algebra which analyses logic mathematically. This Boolean algebra provided a basic logic for operation on binary numbers 0, 1. Since computers are based on binary system, this branch of mathematics is found to be exactly useful for the internal working of various computers. Boolean Algebra is an algebraic structure defined by a set of elements B, together with two operations, + and . satisfying the following axioms (Hunington postulates).
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42. Boolean Algebra Revisited - Page 1
The system of mathematics which he described in his book has become known as Boolean algebra. Boole was a self taught mathematician who discovered the power
http://users.senet.com.au/~dwsmith/boolean.htm
##### Digital Logic Systems
David N. Warren-Smith, CPEng. South Australia Boolean Algebra revisited - Page 1 An Introductory but fresh look at Boolean Algebra Buy the new edition of the book (November 2006) You might prefer to read these pages in the form of a printed book. More convenient to read and better layed out and with improved diagrams , plus additional material . A second edition of this book is now available. I am no longer offering the first edition. This new edition contains a comprehensive chapter on synchronous circuit design methods and techniques. Most digital applications use synchronous circuitry. See the end of the page for details of the book.
##### Introduction
George Boole made major contributions to the development of mathematical logic and published a book The Mathematical Analysis of Logic in 1847. The system of mathematics which he described in his book has become known as Boolean algebra. Boole was a self taught mathematician who discovered the power of mathematics early in life and became a leading figure in mathematical circles. Boolean algebra became a systematic method of dealing with symbolic logic and a much used method of arguing about the fundamentals of mathematics. In 1938 Claude Shannon published an extract from his Masters thesis entitled: A Symbolic Analysis of Relay and Switching Circuits . Shannon made use of Boolean algebra to develop a system that described the logical relationships in switching circuits with simplification of these circuits as one objective. Essentially this provided an algebraic method of describing and manipulating switching circuits. In 1938 there were no logic gate circuits. Electromechanical relay circuits used by the Post Office and in control circuits were the main motivation for developing switching circuit theory.

43. Logic Gates And Boolean Algebra
Logic Gates and Boolean Algebra. Created by Mark Mamo and Shane Bauman. The following is a set of resources for a unit on Logic Gates and Boolean Algebra.
http://educ.queensu.ca/~compsci/resources/BoolLogic/titlepage.html
 Logic Gates and Boolean Algebra Created by Mark Mamo and Shane Bauman The following is a set of resources for a unit on Logic Gates and Boolean Algebra. Introduction to Boolean Logic an outline of an activity to get students thinking about situations using Boolean logic. This activity also serves as an introduction to the AND and OR logic gates. Black Box Circuits an interesting hands-on activity that investigates different gate combinations as well as introduces NAND, NOR. XOR and XNOR Summary of Logic Gates a convenient hand-out summarizing the basic logic gates, their Boolean algebra notation and their truth tables Sample Questions on Logic Gates, Circuits and Truth Tables a handout for students to complete to reinforce the ideas of logic gates, circuits, truth tables and the relationships between them Discovering the Rules of Boolean Algebra a series of worksheets to help students discover the rules of Boolean algebra for themselves Simplifying Boolean Expressions a worksheet which helps students to discover the value of simplifying Boolean expressions and the role it plays in designing circuits

 44. Boolean Algebra@Everything2.com Boolean Algebra is a form of algebra for manipulating Boolean expressions. Unlike normal algebra, variables in Boolean algebra are either True or False.http://everything2.com/index.pl?node_id=413665

45. Definition: Boolean Algebra
Boolean algebra n a system of symbolic logic devised by George Boole; used in computers syn Boolean logic, Boolean algebra
http://dict.die.net/boolean algebra/
##### Definition: boolean algebra
Search dictionary for Source: WordNet (r) 1.7 Boolean algebra n : a system of symbolic logic devised by George Boole; used in computers [syn: Boolean logic , Boolean algebra] Source: The Free On-line Dictionary of Computing (2003-OCT-10) Boolean algebra George Boole ) 1. Commonly, and especially in computer science and digital electronics, this term is used to mean two-valued logic . 2. This is in stark contrast with the definition used by pure mathematicians who in the 1960s introduced "Boolean-valued models " into logic precisely because a "Boolean-valued model" is an interpretation of a theory that allows more than two possible truth values! Strangely, a Boolean algebra (in the mathematical sense) is not strictly an algebra , but is in fact a lattice . A Boolean algebra is sometimes defined as a "complemented distributive lattice ". Boole's work which inspired the mathematical definition concerned algebras of set s, involving the operations of intersection, union and complement on sets. Such algebras obey the following identities where the operators ^, V, - and constants 1 and can be thought of either as set intersection, union, complement, universal, empty; or as two-valued logic AND, OR, NOT, TRUE, FALSE; or any other conforming system. a ^ b = b ^ a a V b = b V a (commutative laws) (a ^ b) ^ c = a ^ (b ^ c) (a V b) V c = a V (b V c) (associative laws) a ^ (b V c) = (a ^ b) V (a ^ c) a V (b ^ c) = (a V b) ^ (a V c) (distributive laws) a ^ a = a a V a = a (idempotence laws) a = a -(a ^ b) = (-a) V (-b) -(a V b) = (-a) ^ (-b) (de Morgan's laws) a ^ -a = a V -a = 1 a ^ 1 = a a V = a a ^ = a V 1 = 1 -1 = -0 = 1 There are several common alternative notations for the "-" or

46. Boolean Algebra, My First Experience. Â« Shriphani’s Blog
After returning home, I picked up a book titled ÂAn Unusual AlgebraÂ by I.M. Yaglom. It is an excellent work that introduces Boolean algebra.
http://shriphani.wordpress.com/2007/12/07/boolean-algebra-my-first-experience/
GHOP, mukt.in stickers, sad news from NIPL and new ideas. NIPL URMS
##### December 7, 2007 6 Comments
If we have sets like A, B and C and if we define addition to be union, and multiplication to be intersection, then we have the following properties associated with the operations addition and multiplication: 1. Commutative property: A + B = B + A or A + C = C + A or B + C = C + B AB = BA or AC = CA or BC = CB 2. Associative propery: (A + B) + C = A + (B + C) (AB)C = A(BC) 3. Distributive property: (A + B)C = AC + BC (A + C)(B + C) = AB + C 4. Idempotent property: AA = A, BB = B and CC = C A + A = A, B + B = B and C + C = C I was then musing that those properties that we stated for sets form the peoperties for operations in Boolean algebra. However I did find a catch in that. We have what is known as the Identity element for addition and multiplication, and 1 respectively. But there is no such set X such that X + A = A or XA = A. If there were such a set, it would be the superset of every set. There you go. I need to learn a bit more. I will be posting more about this book here. Till then, goodbye

47. Wiki Boolean Algebra (logic)
Boolean algebra (or Boolean logic) is a logical calculus of truth values, developed by George Boole. It resembles the algebra of real numbers as taught in
http://wapedia.mobi/en/Boolean_algebra_(logic)
 Wiki: Boolean algebra (logic) Contents: 1. Values 1. 1. Conventions 1. 2. Applications 2. Operations ... Wapedia: For Wikipedia on mobile phones

48. Boolean Algebra - Wikipedia
The most important Boolean algebra, and the one originally described by George Boole, has only two elements, 0 and 1, and is defined by the rules
http://nostalgia.wikipedia.org/wiki/Boolean_algebra