Handbook of Boolean algebras / edited by J. Donald Monk, with the cooperation of Robert Bonnet.
Editor: Amsterdam : North-Holland, 1989Descripción: 3 v. (xix, 1367 p.) : il. ; 25 cmISBN: 0444872914 (set); 044470261X (v. 1); 0444871527 (v. 2); 0444871535 (v. 3)Tema(s): Algebra, Boolean -- Handbooks, manuals, etcOtra clasificación: 06-02 (03B25 03C65 03D35 03G05 06Exx)Contents of Volume [1] Introduction to the Handbook v Contents of the Handbook xi Part I. General Theory of Boolean Algebras, by Sabine Koppelberg [1] Acknowledgements [2] Introduction to Part 1 [3] Chapter 1. Elementary arithmetic [5] Introduction [7] 1. Examples and arithmetic of Boolean algebras [7] 1.1. Definitions and notation [7] 1.2. Algebras of sets [9] 1.3. Lindenbaum—Tarski algebras [11] 1.4. The duality principle [13] 1.5. Arithmetic of Boolean algebras. Connection with lattices [13] 1.6. Connection with Boolean rings [18] 1.7. Infinite operations [20] 1.8. Boolean algebras of projections [23] 1.9. Regular open algebras [25] Exercises [27] 2. Atoms, ultrafilters, and Stone’s theorem [28] 2.1. Atoms [28] 2.2. Ultrafilters and Stone’s theorem [31] 2.3. Arithmetic revisited [34] 2.4. The Rasiowa-Sikorski lemma [35] Exercises [37] 3. Relativization and disjointness [38] 3.1. Relative algebras and pairwise disjoint' families [39] 3.2. Attainment of cellularity: the Erdos-Tarski theorem [41] 3.3. Disjoint refinements: the Balcar-Vojtas theorem [43] Exercises [46] Chapter 2. Algebraic theory [7] Introduction [49] 4. Subalgebras, denseness, and incomparability [50] 4.1. Normal forms [50] 4.2. The completion of a partial order [54] 4.3. The completion of a Boolean algebra [59] 4.4. Irredundance and pairwise incomparable families [61] Exercises [64] 5. Homomorphisms, ideals, and quotients [65] 5.1. Homomorphic extensions [65] 5.2. Sikorski's extension theorem [70] 5.3. Vaught's isomorphism theorem [72] 5.4. Ideals and quotients [74] 5.5. The algebra P(w)/fin [78] 5.6. The number of ultrafilters, filters, and subalgebras [82] Exercises [84] 6. Products [85] 6.1. Product decompositions and partitions [86] 6.2. Hanfs example [88] Exercises [91] Chapter 3. Topological duality [93] Introduction [95] 7. Boolean algebras and Boolean spaces [95] 7.1. Boolean spaces [96] 7.2. The topological version of Stone’s theorem [99] 7.3. Dual properties of A and Ult A [102] Exercises [106] 8. Homomorphisms and continuous maps [106] 8.1. Duality of homomorphisms and continuous maps [107] 8.2. Subalgebras and Boolean equivalence relations [109] 8.3. Product algebras and compactifications [111] 8.4. The sheaf representation of a Boolean algebra over a subalgebra [116] Exercises [125] Chapter 4. Free constructions [127] Introduction [129] 9. Free Boolean algebras [129] 9.1. General facts [130] 9.2. Algebraic and combinatorial properties of free algebras [134] Exercises [139] 10, Independence and the number of ideals [139] 10.1. Independence and chain conditions [140] 10.2. The number of ideals of a Boolean algebra [145] 10.3. A characterization of independence [153] Exercises [157] 11. Free products [157] 11.1. Free products [158] 11.2. Homogeneity, chain conditions, and independence in free products [164] 11.3. Amalgamated free products [168] Exercises [172] Chapter 5. Infinite operations [173] Introduction [175] 12. k-complete algebras [175] 12.1. The countable separation property [176] 12.2. A Schroder-Bernstein theorem [179] 12.3. The Loomis-Sikorski theorem [181] 12.4. Amalgamated free products and injectivity in the category of x-complete Boolean algebras [185] Exercises [189] 13. Complete algebras [190] 13.1. Countably generated complete algebras [190] 13.2. The Balcar-Franek theorem [196] 13.3. Two applications of the Balcar-Franek theorem [204] 13.4. Automorphisms of complete algebras: Frolik’s theorem [207] Exercises [211] 14. Distributive laws [212] 14.1. Definitions and examples [213] 14.2. Equivalences to distributivity [216] 14.3. Distributivity and representability [221] 14.4. Three-parameter distributivity [223] 14.5. Distributive laws in regular open algebras of trees [228] 14.6. Weak distributivity [232] Exercises [236] Chapter 6. Special classes of Boolean algebras [239] Introduction [241] 15. Interval algebras [241] 15.1. Characterization of interval algebras and their dual spaces [242] 15.2. Closure properties of interval algebras [246] 15.3. Retractive algebras [250] 15.4. Chains and antichains in subalgebras of interval algebras [252] Exercises [254] 16. Tree algebras [254] 16.1. Normal forms [255] 16.2. Basic facts on tree algebras [260] 16.3. A construction of rigid Boolean algebras [263] 16.4. Closure properties of tree algebras [265] Exercises [270] 17. Superatomic algebras [271] 17.1. Characterizations of superatomicity [272] 17.2. The Cantor-Bendixson invariants [275] 17.3. Cardinal sequences [277] Exercises [283] Chapter 7. Metamathematics [285] Introduction [287] 18. Decidability of the first order theory of Boolean algebras [287] 18.1. The elementary invariants [288] 18.2. Elementary equivalence of Boolean algebras [293] 18.3. The decidability proof [297] Exercises [299] 19. Undecidability of the first order theory of Boolean algebras with a distinguished subalgebra [299] 19.1. The method of semantical embeddings [300] 19.2. Undecidability of Th (BP*) [303] Exercises [307] References to Part I [309] Index of notation, Volume 1 312a Index, Volume 1 312f
Contents of Volume [2] Introduction to the Handbook v Contents of the Handbook xi Part II. Topics in the theory of Boolean algebras [313] Section A. Arithmetical properties of Boolean algebras [315] Chapter 8. Distributive laws, by Thomas Jech [317] References [331] Chapter 9. Disjoint refinement, by Bohuslav Balcar and Petr Simon [333] 0. Introduction [335] 1. The disjoint refinement property in Boolean algebras [337] 2. The disjoint refinement property of centred systems in Boolean algebras [344] 3. Non-distributivity of P(w) /fin [349] 4. Refinements by countable sets [356] 5. The algebra P(k)/[k]<k; non-distributivity and decomposability [371] References [384] Section B. Algebraic properties of Boolean algebras [387] Chapter 10. Subalgebras, by Robert Bonnet [389] 0. Introduction [391] 1. Characterization of the lattice of subalgebras of a Boolean algebra [393] 2. Complementation and retractiveness in Sub(B) [400] 3. Quasi-complements [408] 4. Congruences on the lattice of subalgebras [414] References [415] Chapter 11. Cardinal functions on Boolean spaces, by Eric K, van Douwen [417] 1. Introduction [419] 2. Conventions [420] 3. A little bit of topology [420] 4. New cardinal functions from old [421] 5. Topological cardinal functions: c, d, L, t, w, π, Xc, πx [422] 6. Basic results [428] 7. Variations of independence [432] II 8. π-weight and π-character [438] 9. Character and cardinality, independence and π-character [443] 10. Getting small dense subsets by killing witnesses [447] 11. Weakly countably complete algebras [451] 12. Cofinality of Boolean algebras and some other small cardinal functions [458] 13. Survey of results [463] 14. The free BA on k generators [464] References and mathematicians mentioned [466] Chapter 12. The number of Boolean algebras, by J. Donald Monk [469] 0. Introduction [471] 1. Simple constructions [472] 2. Construction of complicated Boolean algebras [482] References [489] Chapter 13. Endomorphisms of Boolean algebras, by J. Donald Monk [491] 0. Introduction [493] 1. Reconstruction [493] 2. Number of endomorphisms [497] 3. Endo-rigid algebras [498] 4. Hopfian Boolean algebras [508] Problems [515] References [516] Chapter 14. Automorphism groups, by J. Donald Monk [517] 0. Introduction [519] 1. General properties [519] 2. Galois theory of simple extensions [528] 3. Galois theory of finite extensions [533] 4. The size of automorphism groups [539] References [545] Chapter 15. On the reconstruction of Boolean algebras from their automorphism groups, by Matatyahu Rubin [547] 1. Introduction [549] 2. The method [552] 3. Faithfulness in the class of complete Boolean algebras [554] 4. Faithfulness of incomplete Boolean algebras [574] 5. Countable Boolean algebras [586] 6. Faithfulness of measure algebras [591] References [605] Chapter 16. Embeddings and automorphisms, by Petr Stepánek [607] 0. Introduction [609] 1. Rigid complete Boolean algebras [610] 2. Embeddings into complete rigid algebras [620] 3. Embeddings into the center of a Boolean algebra [624] 4. Boolean algebras with no rigid or homogeneous factors [629] 5. Embeddings into algebras with a trivial center [633] References [635] Chapter 17. Rigid Boolean algebras, by Mohamed Bekkali and Robert Bonnet [637] 0. Introduction [639] 1. Basic concepts concerning orderings and trees [640] 2. The Jonsson construction of a rigid algebra [643] 3. Bonnet’s construction of mono-rigid interval algebras [646] 4. Todorcevic’s construction of many mono-rigid interval algebras [655] 5. Jech’s construction of simple complete algebras [664] 6. Odds and ends on rigid algebras [674] References [676] Chapter 18. Homogeneous Boolean algebras, by Petr Stepanek and Matatyahu Rubin [679] 0. Introduction [681] 1. Homogeneous algebras [681] 2. Weakly homogeneous algebras [683] 3. k -universal homogeneous algebras [685] 4. Complete weakly homogeneous algebras [687] 5. Results and problems concerning the simplicity of automorphism groups of homogeneous BAs [694] 6. Stronger forms of homogeneity [712] References [714] Index of notation, Volume 2 716a Index, Volume 2 716j
Contents of Volume [3] Introduction to the Handbook Contents of the Handbook xi Part II. Topics in the theory of Boolean algebras (continued) [313] Section C. Special classes of Boolean algebras [717] Chapter 19. Superatomic Boolean algebras, by Judy Roitman [719] Introduction [721] Preliminaries [722] Odds and ends [724] Thin-tall Boolean algebras [727] No big sBAs [731] More negative results [733] A very thin thick sBA [735] Any countable group can be G(B) [737] References [739] Chapter 20. Projective Boolean algebras, by Sabine Koppefberg [741] Introduction [743] Elementary results [744] Characterizations of projective algebras [751] Characters of ultrafilters [757] The number of projective Boolean algebras [763] References [772] Chapter 21. Countable Boolean algebras, by R.S. Pierce [775] Introduction [777] Invariants [777] Algebras of isomorphism types [809] Special classes of algebras [847] References [875] Chapter 22. Measure algebras, by David H. Fremlin [877] Introduction [879] Measure theory [880] Measure algebras [888] Maharam’s theorem [907] Liftings [928] Which algebras are measurable? [940] Cardinal functions [956] Envoi: Atomlessly-measurable cardinals [973] References [976] Section D. Logical questions [981] Chapter 23. Decidable extensions of the theory of Boolean algebras, by Martin Weese [983] Introduction [985] Describing the languages [986] The monadic theory of countable linear orders and its application to the theory of Boolean algebras [993] The theories Thu(BA) and ThQd(BA) [1002] Ramsey quantifiers and sequence quantifiers [1010] The theory of Boolean algebras with cardinality quantifiers [1021] Residually small discriminator varieties [1034] Boolean algebras with a distinguished finite automorphism group [1050] Boolean pairs [1055] References [1065] Chapter 24. Undecidable extensions of the theory of Boolean algebras, by Martin Weese [1067] Introduction [1069] Boolean algebras in weak second-order logic and second-order logic [1070] Boolean algebras in a logic with the Härtig quantifier [1072] Boolean algebras in a logic with the Malitz quantifier [1074] Boolean algebras in stationary logic [1076] Boolean algebras with a distinguished group of automorphisms [1079] Single Boolean algebras with a distinguished ideal [1081] Boolean algebras in a logic with quantification over ideals [1083] Some applications [1088] References [1095] Chapter 25. Recursive Boolean algebras, by J.B. Remmel [1097] Introduction [1099] Preliminaries [1101] Equivalent characterizations of recursive, r.e., and arithmetic BAs [1108] Recursive Boolean algebras with highly effective presentations [1112] Recursive Boolean algebras with minimally effective presentations [1125] Recursive isomorphism types of Rec. BAs [1140] The lattices of r.e. subalgebras and r.e. ideals of a Rec. BA [1151] Recursive automorphisms of Rec. BAs [1159] References [1162] Chapter 26. Lindenbaum-Tarski algebras, by Dale Myers [1167] Introduction [1169] History [1169] Sentence algebras and model spaces [1170] Model maps [1171] Duality [1173] Repetition and Cantor-Bernstein [1175] Language isomorphisms [1176] Measures [1178] Rank diagrams [1179] Interval algebras and cut spaces [1183] Finite monadic languages [1185] Factor measures [1187] Measure monoids [1187] Orbits [1188] Primitive spaces and orbit diagrams [1190] Miscellaneous [1191] Table of sentence algebras [1193] References [1193] Chapter 27. Boolean-valued models, by Thomas Jech [1197] Appendix on set theory, by J. Donald Monk [1213] Introduction [1215] Cardinal arithmetic [1215] Two lemmas on the unit interval [1218] Almost-disjoint sets [1221] Independent sets [1221] Stationary sets [1222] ∆-systems [1227] The partition calculus [1228] Hajnal’s free set theorem [1231] References [1233] Chart of topological duality [1235] Appendix on general topology, by Bohuslav Balcar and Petr Simon [1239] Introduction [1241] Basics [1241] Separation axioms [1245] Compactness [1247] The Cech-Stone compactification [1250] Extremally disconnected and Gleason spaces [1253] K-Parovicenko spaces [1257] F-spaces [1261] Cardinal invariants [1265] References [1266] Bibliography [1269] General [1269] Elementary [1299] Functional analysis [1309] Logic [1311] Measure algebras [1317] Recursive BAs [1327] Set theory and BAs [1329] Topology and BAs [1332] Topological BAs [1340] Index of notation, Volume 3 [1343] Index, Volume 3 [1351]
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Vol. 1 por Sabine Koppelberg.
Incluye referencias bibliográficas e índices.
Contents of Volume [1] --
Introduction to the Handbook v --
Contents of the Handbook xi --
Part I. General Theory of Boolean Algebras, by Sabine Koppelberg [1] --
Acknowledgements [2] --
Introduction to Part 1 [3] --
Chapter 1. Elementary arithmetic [5] --
Introduction [7] --
1. Examples and arithmetic of Boolean algebras [7] --
1.1. Definitions and notation [7] --
1.2. Algebras of sets [9] --
1.3. Lindenbaum—Tarski algebras [11] --
1.4. The duality principle [13] --
1.5. Arithmetic of Boolean algebras. Connection with lattices [13] --
1.6. Connection with Boolean rings [18] --
1.7. Infinite operations [20] --
1.8. Boolean algebras of projections [23] --
1.9. Regular open algebras [25] --
Exercises [27] --
2. Atoms, ultrafilters, and Stone’s theorem [28] --
2.1. Atoms [28] --
2.2. Ultrafilters and Stone’s theorem [31] --
2.3. Arithmetic revisited [34] --
2.4. The Rasiowa-Sikorski lemma [35] --
Exercises [37] --
3. Relativization and disjointness [38] --
3.1. Relative algebras and pairwise disjoint' families [39] --
3.2. Attainment of cellularity: the Erdos-Tarski theorem [41] --
3.3. Disjoint refinements: the Balcar-Vojtas theorem [43] --
Exercises [46] --
Chapter 2. Algebraic theory [7] --
Introduction [49] --
4. Subalgebras, denseness, and incomparability [50] --
4.1. Normal forms [50] --
4.2. The completion of a partial order [54] --
4.3. The completion of a Boolean algebra [59] --
4.4. Irredundance and pairwise incomparable families [61] --
Exercises [64] --
5. Homomorphisms, ideals, and quotients [65] --
5.1. Homomorphic extensions [65] --
5.2. Sikorski's extension theorem [70] --
5.3. Vaught's isomorphism theorem [72] --
5.4. Ideals and quotients [74] --
5.5. The algebra P(w)/fin [78] --
5.6. The number of ultrafilters, filters, and subalgebras [82] --
Exercises [84] --
6. Products [85] --
6.1. Product decompositions and partitions [86] --
6.2. Hanfs example [88] --
Exercises [91] --
Chapter 3. Topological duality [93] --
Introduction [95] --
7. Boolean algebras and Boolean spaces [95] --
7.1. Boolean spaces [96] --
7.2. The topological version of Stone’s theorem [99] --
7.3. Dual properties of A and Ult A [102] --
Exercises [106] --
8. Homomorphisms and continuous maps [106] --
8.1. Duality of homomorphisms and continuous maps [107] --
8.2. Subalgebras and Boolean equivalence relations [109] --
8.3. Product algebras and compactifications [111] --
8.4. The sheaf representation of a Boolean algebra over a subalgebra [116] --
Exercises [125] --
Chapter 4. Free constructions [127] --
Introduction [129] --
9. Free Boolean algebras [129] --
9.1. General facts [130] --
9.2. Algebraic and combinatorial properties of free algebras [134] --
Exercises [139] --
10, Independence and the number of ideals [139] --
10.1. Independence and chain conditions [140] --
10.2. The number of ideals of a Boolean algebra [145] --
10.3. A characterization of independence [153] --
Exercises [157] --
11. Free products [157] --
11.1. Free products [158] --
11.2. Homogeneity, chain conditions, and independence in free products [164] --
11.3. Amalgamated free products [168] --
Exercises [172] --
Chapter 5. Infinite operations [173] --
Introduction [175] --
12. k-complete algebras [175] --
12.1. The countable separation property [176] --
12.2. A Schroder-Bernstein theorem [179] --
12.3. The Loomis-Sikorski theorem [181] --
12.4. Amalgamated free products and injectivity in the category of --
x-complete Boolean algebras [185] --
Exercises [189] --
13. Complete algebras [190] --
13.1. Countably generated complete algebras [190] --
13.2. The Balcar-Franek theorem [196] --
13.3. Two applications of the Balcar-Franek theorem [204] --
13.4. Automorphisms of complete algebras: Frolik’s theorem [207] --
Exercises [211] --
14. Distributive laws [212] --
14.1. Definitions and examples [213] --
14.2. Equivalences to distributivity [216] --
14.3. Distributivity and representability [221] --
14.4. Three-parameter distributivity [223] --
14.5. Distributive laws in regular open algebras of trees [228] --
14.6. Weak distributivity [232] --
Exercises [236] --
Chapter 6. Special classes of Boolean algebras [239] --
Introduction [241] --
15. Interval algebras [241] --
15.1. Characterization of interval algebras and their dual spaces [242] --
15.2. Closure properties of interval algebras [246] --
15.3. Retractive algebras [250] --
15.4. Chains and antichains in subalgebras of interval algebras [252] --
Exercises [254] --
16. Tree algebras [254] --
16.1. Normal forms [255] --
16.2. Basic facts on tree algebras [260] --
16.3. A construction of rigid Boolean algebras [263] --
16.4. Closure properties of tree algebras [265] --
Exercises [270] --
17. Superatomic algebras [271] --
17.1. Characterizations of superatomicity [272] --
17.2. The Cantor-Bendixson invariants [275] --
17.3. Cardinal sequences [277] --
Exercises [283] --
Chapter 7. Metamathematics [285] --
Introduction [287] --
18. Decidability of the first order theory of Boolean algebras [287] --
18.1. The elementary invariants [288] --
18.2. Elementary equivalence of Boolean algebras [293] --
18.3. The decidability proof [297] --
Exercises [299] --
19. Undecidability of the first order theory of Boolean algebras with a distinguished subalgebra [299] --
19.1. The method of semantical embeddings [300] --
19.2. Undecidability of Th (BP*) [303] --
Exercises [307] --
References to Part I [309] --
Index of notation, Volume 1 312a --
Index, Volume 1 312f --
Contents of Volume [2] --
Introduction to the Handbook v --
Contents of the Handbook xi --
Part II. Topics in the theory of Boolean algebras [313] --
Section A. Arithmetical properties of Boolean algebras [315] --
Chapter 8. Distributive laws, by Thomas Jech [317] --
References [331] --
Chapter 9. Disjoint refinement, by Bohuslav Balcar and Petr Simon [333] --
0. Introduction [335] --
1. The disjoint refinement property in Boolean algebras [337] --
2. The disjoint refinement property of centred systems in Boolean algebras [344] --
3. Non-distributivity of P(w) /fin [349] --
4. Refinements by countable sets [356] --
5. The algebra P(k)/[k]<k; non-distributivity and decomposability [371] --
References [384] --
Section B. Algebraic properties of Boolean algebras [387] --
Chapter 10. Subalgebras, by Robert Bonnet [389] --
0. Introduction [391] --
1. Characterization of the lattice of subalgebras of a Boolean algebra [393] --
2. Complementation and retractiveness in Sub(B) [400] --
3. Quasi-complements [408] --
4. Congruences on the lattice of subalgebras [414] --
References [415] --
Chapter 11. Cardinal functions on Boolean spaces, by Eric K, van Douwen [417] --
1. Introduction [419] --
2. Conventions [420] --
3. A little bit of topology [420] --
4. New cardinal functions from old [421] --
5. Topological cardinal functions: c, d, L, t, w, π, Xc, πx [422] --
6. Basic results [428] --
7. Variations of independence [432] --
II 8. π-weight and π-character [438] --
9. Character and cardinality, independence and π-character [443] --
10. Getting small dense subsets by killing witnesses [447] --
11. Weakly countably complete algebras [451] --
12. Cofinality of Boolean algebras and some other small cardinal functions [458] --
13. Survey of results [463] --
14. The free BA on k generators [464] --
References and mathematicians mentioned [466] --
Chapter 12. The number of Boolean algebras, by J. Donald Monk [469] --
0. Introduction [471] --
1. Simple constructions [472] --
2. Construction of complicated Boolean algebras [482] --
References [489] --
Chapter 13. Endomorphisms of Boolean algebras, by J. Donald Monk [491] --
0. Introduction [493] --
1. Reconstruction [493] --
2. Number of endomorphisms [497] --
3. Endo-rigid algebras [498] --
4. Hopfian Boolean algebras [508] --
Problems [515] --
References [516] --
Chapter 14. Automorphism groups, by J. Donald Monk [517] --
0. Introduction [519] --
1. General properties [519] --
2. Galois theory of simple extensions [528] --
3. Galois theory of finite extensions [533] --
4. The size of automorphism groups [539] --
References [545] --
Chapter 15. On the reconstruction of Boolean algebras from their automorphism groups, by Matatyahu Rubin [547] --
1. Introduction [549] --
2. The method [552] --
3. Faithfulness in the class of complete Boolean algebras [554] --
4. Faithfulness of incomplete Boolean algebras [574] --
5. Countable Boolean algebras [586] --
6. Faithfulness of measure algebras [591] --
References [605] --
Chapter 16. Embeddings and automorphisms, by Petr Stepánek [607] --
0. Introduction [609] --
1. Rigid complete Boolean algebras [610] --
2. Embeddings into complete rigid algebras [620] --
3. Embeddings into the center of a Boolean algebra [624] --
4. Boolean algebras with no rigid or homogeneous factors [629] --
5. Embeddings into algebras with a trivial center [633] --
References [635] --
Chapter 17. Rigid Boolean algebras, by Mohamed Bekkali and Robert Bonnet [637] --
0. Introduction [639] --
1. Basic concepts concerning orderings and trees [640] --
2. The Jonsson construction of a rigid algebra [643] --
3. Bonnet’s construction of mono-rigid interval algebras [646] --
4. Todorcevic’s construction of many mono-rigid interval algebras [655] --
5. Jech’s construction of simple complete algebras [664] --
6. Odds and ends on rigid algebras [674] --
References [676] --
Chapter 18. Homogeneous Boolean algebras, by Petr Stepanek and Matatyahu Rubin [679] --
0. Introduction [681] --
1. Homogeneous algebras [681] --
2. Weakly homogeneous algebras [683] --
3. k -universal homogeneous algebras [685] --
4. Complete weakly homogeneous algebras [687] --
5. Results and problems concerning the simplicity of automorphism groups of homogeneous BAs [694] --
6. Stronger forms of homogeneity [712] --
References [714] --
Index of notation, Volume 2 716a --
Index, Volume 2 716j --
Contents of Volume [3] --
Introduction to the Handbook --
Contents of the Handbook xi --
Part II. Topics in the theory of Boolean algebras (continued) [313] --
Section C. Special classes of Boolean algebras [717] --
Chapter 19. Superatomic Boolean algebras, by Judy Roitman [719] --
Introduction [721] --
Preliminaries [722] --
Odds and ends [724] --
Thin-tall Boolean algebras [727] --
No big sBAs [731] --
More negative results [733] --
A very thin thick sBA [735] --
Any countable group can be G(B) [737] --
References [739] --
Chapter 20. Projective Boolean algebras, by Sabine Koppefberg [741] --
Introduction [743] --
Elementary results [744] --
Characterizations of projective algebras [751] --
Characters of ultrafilters [757] --
The number of projective Boolean algebras [763] --
References [772] --
Chapter 21. Countable Boolean algebras, by R.S. Pierce [775] --
Introduction [777] --
Invariants [777] --
Algebras of isomorphism types [809] --
Special classes of algebras [847] --
References [875] --
Chapter 22. Measure algebras, by David H. Fremlin [877] --
Introduction [879] --
Measure theory [880] --
Measure algebras [888] --
Maharam’s theorem [907] --
Liftings [928] --
Which algebras are measurable? [940] --
Cardinal functions [956] --
Envoi: Atomlessly-measurable cardinals [973] --
References [976] --
Section D. Logical questions [981] --
Chapter 23. Decidable extensions of the theory of Boolean algebras, by --
Martin Weese [983] --
Introduction [985] --
Describing the languages [986] --
The monadic theory of countable linear orders and its application to the theory of Boolean algebras [993] --
The theories Thu(BA) and ThQd(BA) [1002] --
Ramsey quantifiers and sequence quantifiers [1010] --
The theory of Boolean algebras with cardinality quantifiers [1021] --
Residually small discriminator varieties [1034] --
Boolean algebras with a distinguished finite automorphism group [1050] --
Boolean pairs [1055] --
References [1065] --
Chapter 24. Undecidable extensions of the theory of Boolean algebras, by Martin Weese [1067] --
Introduction [1069] --
Boolean algebras in weak second-order logic and second-order logic [1070] --
Boolean algebras in a logic with the Härtig quantifier [1072] --
Boolean algebras in a logic with the Malitz quantifier [1074] --
Boolean algebras in stationary logic [1076] --
Boolean algebras with a distinguished group of automorphisms [1079] --
Single Boolean algebras with a distinguished ideal [1081] --
Boolean algebras in a logic with quantification over ideals [1083] --
Some applications [1088] --
References [1095] --
Chapter 25. Recursive Boolean algebras, by J.B. Remmel [1097] --
Introduction [1099] --
Preliminaries [1101] --
Equivalent characterizations of recursive, r.e., and arithmetic BAs [1108] --
Recursive Boolean algebras with highly effective presentations [1112] --
Recursive Boolean algebras with minimally effective presentations [1125] --
Recursive isomorphism types of Rec. BAs [1140] --
The lattices of r.e. subalgebras and r.e. ideals of a Rec. BA [1151] --
Recursive automorphisms of Rec. BAs [1159] --
References [1162] --
Chapter 26. Lindenbaum-Tarski algebras, by Dale Myers [1167] --
Introduction [1169] --
History [1169] --
Sentence algebras and model spaces [1170] --
Model maps [1171] --
Duality [1173] --
Repetition and Cantor-Bernstein [1175] --
Language isomorphisms [1176] --
Measures [1178] --
Rank diagrams [1179] --
Interval algebras and cut spaces [1183] --
Finite monadic languages [1185] --
Factor measures [1187] --
Measure monoids [1187] --
Orbits [1188] --
Primitive spaces and orbit diagrams [1190] --
Miscellaneous [1191] --
Table of sentence algebras [1193] --
References [1193] --
Chapter 27. Boolean-valued models, by Thomas Jech [1197] --
Appendix on set theory, by J. Donald Monk [1213] --
Introduction [1215] --
Cardinal arithmetic [1215] --
Two lemmas on the unit interval [1218] --
Almost-disjoint sets [1221] --
Independent sets [1221] --
Stationary sets [1222] --
∆-systems [1227] --
The partition calculus [1228] --
Hajnal’s free set theorem [1231] --
References [1233] --
Chart of topological duality [1235] --
Appendix on general topology, by Bohuslav Balcar and Petr Simon [1239] --
Introduction [1241] --
Basics [1241] --
Separation axioms [1245] --
Compactness [1247] --
The Cech-Stone compactification [1250] --
Extremally disconnected and Gleason spaces [1253] --
K-Parovicenko spaces [1257] --
F-spaces [1261] --
Cardinal invariants [1265] --
References [1266] --
Bibliography [1269] --
General [1269] --
Elementary [1299] --
Functional analysis [1309] --
Logic [1311] --
Measure algebras [1317] --
Recursive BAs [1327] --
Set theory and BAs [1329] --
Topology and BAs [1332] --
Topological BAs [1340] --
Index of notation, Volume 3 [1343] --
Index, Volume 3 [1351] --
MR, 90k:06002 (v. 1)
MR, 90k:06003 (v. 2)
MR, 90k:06004 (v. 3)
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