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Handbook of Boolean algebras / edited by J. Donald Monk, with the cooperation of Robert Bonnet.

Colaborador(es): Monk, J. Donald (James Donald), 1930- [edt] | Bonnet, Robert | Koppelberg, SabineEditor: 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)
Contenidos:
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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Libros Libros Instituto de Matemática, CONICET-UNS
06 H236 (Browse shelf) Vol. 1 Available A-6358

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Libros Libros Instituto de Matemática, CONICET-UNS
06 H236 (Browse shelf) Vol. 2 Available A-6359
Libros Libros Instituto de Matemática, CONICET-UNS
06 H236 (Browse shelf) Vol. 3 Available A-6360

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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