Analytic quotients theory of liftings for quotients over analytic ideals on the integers

Memoirs of the American Mathematical Society

Volumn , Issue 702, 2000, Pages 1-171

  Farah, Ilijas ^a  

^a NONE

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0034311439     PISSN: 00659266     EISSN: None     Source Type: Journal    
DOI: 10.1090/memo/0702     Document Type: Article

References (146)

1. 1-dense real order types. Annals of Pure and Applied Logic, 29:123-206, 1985.
2. [2] T. Bartoszynski. Invariants of measure and category. In M. Foreman and A. Kanamori, editors, Handbook of Set Theory. to appear.
3. 1-dense sets of reals can be isomorphic. Fundamenta Mathematicae, 79:101-106, 1973.
4. [4] J.E. Baumgartner. Chains and antichains in P(ω). The Journal of Symbolic Logic, 45:85-92, 1980.
5. [5] M. Bekkali. Topics in Set Theory, volume 1476 of Lecture Notes in Mathematics. Springer-Verlag, 1991.
6. [6] M. Bell. an email of October 1997.
7. [7] M. Bell. Two Boolen algebras with extreme cellular and compactness properties. Canadian Journal of Mathematics, XXXV:824-838, 1983.
8. [8] A. Blass. Near coherence of filters II: Applications to operator ideals, the Stone-čech remainder of a half-line, order ideals of sequences, and slenderness of groups. Transactions of the American Mathematical Society, 300:557-581, 1987.
9. [9] A. Blass and C. Laflamme. Consistency results about filters and the number of inequivalent growth types. The Journal of Symbolic Logic, 54:50-56, 1989.
10. [10] M. Burke. Liftings for Lebesgue measure. In Set Theory of the reals, volume 6 of Israel Mathematical Conference Proceedings, pages 119-150. 1993.
11. [11] P. Casevitz. Dichotomies pour les espaces de suites reelles. PhD thesis, Universite Paris 6, 1999.
12. [12] C.C. Chang and H.J. Keisler. Model Theory. North-Holland, 1973.
13. [13] J.P.R. Christensen. Some results with relation to the control measure problem. In R.M. Aron and S. Dineen, editors, Vector space measures and applications II, volume 645 of Lecture Notes in Mathematics, pages 27-34. Springer, 1978.
14. [14] W.W. Comfort and S. Negrepontis. Theory of ultrafilters. Springer, 1974.
15. [15] H.G. Dales and W.H. Woodin. An Introduction to Independence for Analysts, volume 115 of London Mathematical Society Lecture Note Series. Cambridge University Press, 1987.
16. [16] J. Diestel and J.J. Uhl, Jr. Vector measures. Amer. Math. Soc. Math. Surveys, Number 15, 1977.
17. [17] E.K. van Douwen. Prime mappings, number of factors and binary operations, volume 199 of Dissertationes Mathematicae. Warszawa, 1981.
18. [18] E.K. van Douwen. The automorphism group of P(ω)/Fin need not be simple. Topology and its Applications, pages 97-104, 1990.
19. [19] E.K. van Douwen and J. van Mill. The homeomorphism extension theorem for βω \ ω. In S. Andima et al., editors, Papers on general topology and applications, volume 704 of Ann. New York Acad. Sci., pages 345-350. 1993.
20. [20] E.K. van Douwen, D. Monk, and M. Rubin. Some questions about Boolean algebras. Algebra Universalis, 11:220-243, 1980.
21. [21] A. Dow. Extending real-valued functions in βκ. Fundamenta Mathematicae, 152:21-41, 1997.
22. [22] A. Dow and I. Farah. Is P(ω) a subalgebra? in preparation, 2001.
23. [23] A. Dow and K.P. Hart. ω* has (almost) no continuous images. Israel Journal of Mathematics, 109:29-39, 1999.
24. [24] A. Dow and K.P. Hart. The measure algebra does not always embed. Fundamenta Mathematicae, 163:163-176, 2000.
25. [25] A. Dow, P. Simon, and J.E. Vaughan. Strong homology and the Proper Forcing Axiom. Proceedings of the American Mathematical Society, 106 (3):821-828, 1989.
26. [26] A. Ehrenfeucht, J. Kahn, R. Maddux, and J. Mycielski. On the dependence of functions on their variables. Journal of Combinatorial Theory, ser. A, 33:106-108, 1982.
27. [27] R. Engelking. General Topology. Heldermann, Berlin, 1989.
28. [28] P. Erdös. My Scottish Book "problems". In D. Mauldin, editor, The Scottish Book pages 35-44. Birkhäuser, Boston, 1981.
29. [29] I. Farah. A coherent family of functions. Proceedings of the American Mathematical Society, 124:2845-2852, 1996.
30. ω. Fundamenta Mathematicae, 151:53-95, 1996.
31. [31] I Farah. OCA and towers in P(ℕ)/Fin. Commentationes Mathematicae Universitatis Carolinae, 37:861-866, 1996.
32. [32] I. Farah. Analytic ideals and their quotients. PhD thesis, University of Toronto, 1997.
33. [33] I. Farah. Approximate homomorphisms. Combinatorica, 18:335-348, 1998.
34. [34] I. Farah. Cauchy nets and open colorings. Publ. Inst. Math. (Beograd) (N.S.), 64(78):146-152, 1998. 50th anniversary of the Mathematical Institute, Serbian Academy of Sciences and Arts (Belgrade, 1996).
35. [35] I. Farah. Completely additive liftings. The Bulletin of Symbolic Logic, 4:37-54, 1998.
36. [36] I. Farah. Ideals induced by Tsirelson submeasures. Fundamenta Mathematicae, 159:243-258, 1999.
37. [37] I. Farah. Approximate homomorphisms II: Group homomorphisms. Combinatorica, 20:47-60, 2000.
38. [38] I. Farah. Liftings of homomorphisms between quotient structures and Ulam stability. In S. Buss, P. Hájek, and P. Pudlák, editors, Logic Colloquium '98, volume 13 of Lecture notes in logic, pages 173-196. A.K. Peters, 2000.
39. [39] I. Farah. Basis problem for turbulent actions I: Tsirelson submeasures. In Proceedings of XI Latin American Symposium in Mathematical Logic, Merida, July 1998, Annals of Pure and Applied Logic. to appear.
40. 0-equalities. Proceedings of the London Mathematical Society, 81, to appear.
41. [41] I. Farah. Dimension phenomena associated with βℕ-spaces. Topology and its Applications, to appear.
42. [42] I. Farah. Functions essentially depending on at most one variable. Journal of Combinatorial Theory, Ser. A, to appear.
43. [43] Qi Feng. Homogeneity for open partitions of pairs of reals. Transactions of the American Mathematical Society, 339:659-684, 1993.
44. [44] G.L. Forti. Hyers-Ulam stability of functional equations in several variables. Aequationes Math., 50:143-190, 1995.
45. [45] D.H. Fremlin. Measure algebras. In D. Monk and R. Bonnett, editors, Handbook of Boolean algebras, pages 877-980. North-Holland, Amsterdam, 1989.
46. [46] D.H. Fremlin. The partially ordered sets of measure theory and Turkey's ordering. Note di Matematica, Vol. XI:177-214, 1991.
47. [47] D.H. Fremlin. Notes on FARAH P99. preprint, University of Essex, June 1999.
48. 0. preprint, University of Essex, May 1999.
49. [49] D.H. Fremlin. Problems. University of Essex, version of april 21, 1994.
50. [50] F. Galvin. Letter of August 5, 1995.
51. [51] J. Hadamard. Sur les caracteres de convergence des series a termes positifs et sur les fonctions indefiniment croissantes. Acta Mathematicae, 18:319-336, 1884.
52. [52] K.P. Hart and J. van Mill. Open problems on βω. In J. van Mill and G.M. Reed, editors, Open problems in topology, pages 97-125. North-Holland, 1990.
53. [53] F. Hausdorff. Die Graduierung nach dem Endverlauf. Abhandlungen der Königlich Sächsischen Gesellschaft der Wissenschaften; Mathematisch-Physiche Klasse, 31:296-334, 1909.
54. 1 Mengen. Fundamenta Mathematicae, 26:241-255, 1936.
55. [55] G. Hjorth and A.S. Kechris. New dichotomies for Borel equivalence relations. The Bulletin of Symbolic Logic, 3:329-346, 1997.
56. [56] W. Hurewicz. Relativ Perfecte Teile von Punktmengen und Mengen (A). Fundamenta Mathematicae, 12:78-109, 1928.
57. [57] A. Ionescu Tulcea and C. Ionescu Tulcea. Topics in the theory of lifting. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 48. Springer-Verlag, New York, 1969.
58. [58] S.-A. Jalali-Naini. The monotone subsets of Cantor space, filters and descriptive set theory. PhD thesis, Oxford, 1976.
59. [59] B Jonsson and P. Olin. Almost direct products and saturation. Compositio Mathematicae, 20:125-132, 1968.
60. [60] W. Just. Nowhere dense P-subsets of ω*. Proceedings of the American Mathematical Society, 106:1145-1146, 1989.
61. n. Fundamenta Mathematicae, 132:59-72, 1989.
62. [62] W. Just. Repercussions on a problem of Erdös and Ulam about density ideals. Canadian Journal of Mathematics, 42:902-914, 1990.
63. [63] W. Just. A modification of Shelah's oracle chain condition with applications. Transactions of the American Mathematical Society, 329:325-341, 1992.
64. [64] W. Just. A weak version of AT from OCA. Mathematical Science Research Institute Publications, 26:281-291, 1992.
65. [65] W. Just and A. Krawczyk. On certain Boolean algebras P(ω)/I. Transactions of the American Mathematical Society, 285:411-429, 1984.
66. [66] N.J. Kalton. The Maharam problem. In G. Choquet et al., editors, Séminaire Initiation à ĺ Analyse, 28e Anńee, volume 18, pages 1-13. 1988/89.
67. [67] N.J. Kalton and J.W. Roberts. Uniformly exhaustive submeasures and nearly additive set functions. Transactions of the American Mathematical Society, 278:803-816, 1983.
68. [68] S. Kamo. Almost coinciding families and gaps in P(ω). J. Math. Soc. Japan, 45:357-368, 1993.
69. [69] A. Kanamori. The higher infinite. Springer-Verlag, 1995.
70. [70] V. Kanovei and M. Reeken. New Radon-Nikodym ideals. Mathematika, to appear.
71. [71] V. Kanovei and M. Reeken. On Baire measurable homomorphisms of quotients of the additive group of the reals. Mathematical Logic Quarterly, to appear.
72. [72] V. Kanovei and M. Reeken. On Ulam's problem of stability of non-exact homomorphisms. Proceedings of Moscow Steklov Mathematical Institute, to appear.
73. [73] I. Kaplansky. Normed algebras. Duke Math. Journal, 16:399-418, 1949.
74. [74] A.S. Kechris. Classical descriptive set theory, volume 156 of Graduate texts in mathematics. Springer, 1995.
75. [75] A.S. Kechris. Rigidity properties of Borel ideals on the integers. In 8th Prague symposium on general topology and its relations to modern analysis and algebra, 1996, volume 85 of Topology and its applications, pages 195-205, 1998.
76. [76] A.S. Kechris and A. Louveau. The structure of hypersmooth Borel equivalence relations. Journal of the American Mathematical Society, 10:215-242, 1997.
77. [77] A.S. Kechris, A. Louveau, and W.H. Woodin. The structure of σ-ideals of compact sets. Transactions of the American Mathematical Society, 301:263-288, 1987.
78. [78] H. J. Keisler. Logic with the quantifier "There exists uncountably many". Annals of Mathematical Logic, 1:1-93, 1970.
79. [79] S. Koppelberg. personal communication.
80. [80] K. Kunen. Some comments on box products. In A. Hajnal et al., editors, Infinite and finite sets, Keszthely (Hungary), 1973, volume 10 of Coll. Math. Soc. Jänos Bolyai, pages 1011-1016. North-Holland, Amsterdam, 1975.
81. [81] K. Kunen. (κ, λ*)-gaps under MA. preprint, 1976.
82. [82] K. Kunen. An Introduction to Independence Proofs. North-Holland , 1980.
83. [83] K. Kunen. Where MA first fails. The Journal of Symbolic Logic, 53:429-433, 1988.
84. [84] C. Laflamme. Forcing with filters and complete combinatorics. Annals of pure and applied logic, 42:125-163, 1989.
85. σ filters witn an application to N-sets. Proceedings of the American Mathematical Society, 125:3019-3025, 1997.
86. [86] C. Laflamme and J. Zhu. The Rudin-Blass ordering on ultrafilers. Journal of Symbolic Logic, 63:584-592, 1998.
87. [87] A. Louveau and B. Velickovic. A note on Borel equivalence relations. Proceedings of the American Mathematical Society, 120:255-259, 1994.
88. [88] A. Louveau and B. Velickovic. Analytic ideals and cofinal types. Annals of Pure and Applied Logic, 99:171-195, 1999.
89. [90] S. Mardesic and A. Prasolov. Strong homology is not additive. Transactions of the American Mathematical Society, 307:725-744, 1988.
90. [91] D.A. Martin and R.M. Solovay. Internal Cohen extensions. Annals of Mathematical Logic, 2:143-178, 1970.
91. [92] A.R.D. Mathias. Solution of problems of Choquet and Puritz. In Conference in Mathematical Logic-London '70, volume 255 of Springer Lecture Notes in Mathematics, pages 204-210. 1972.
92. [93] A.R.D. Mathias. A remark on rare filters. In A. Hajnal et al., editors, Infinite and finite sets, Vol. III, volume 10 of Coll. Math. Soc. Jänos Bolyai, pages 1095-1097. North Holland, 1975.
93. σ-ideals. preprint, 199?
94. 1-gaps in the Boolean algebra P(ω)/I. Fundamenta Mathematicae, 138:103-111, 1991.
95. σ-ideals. preprint, 1996.
96. [97] J. van Mill. An introduction to βω. In K. Kunen and J. Vaughan, editors, Handbook of Set-theoretic topology, pages 503-560. North-Holland, 1984.
97. [98] J.D. Monk and R.M. Solovay. On the number of complete Boolean algebras. Algebra Universalis, 2:365-368, 1972.
98. [99] J.T. Moore. Linearly fibered Souslinean space under MA. Topology Proceedings, 24, Spring 1999.
99. 0. preprint, UCLA, 1997.
100. [101] I.I. Parovičenko. A universal bicompact of weight א. Soviet Mathematics Doklady, 4:592-592, 1963.
101. [102] R.S. Pierce. Countable Boolean algebras. In J.D. Monk and R. Bonnett, editors, Handbook of Boolean algebras, volume 3, pages 775-886. North-Holland, Amsterdam, 1989.
102. [103] F. Rothberger. Sur la families indenombrables de suites de nombres naturels et les problemes concernant la propriete C. Proc. Cambridge Phil. Soc., 37:109-126, 1941.
103. [104] W. Rudin. Homogeneity problems in the theory of čech compactifications. Duke mathematics Journal, 23:409-419, 1956.
104. ωω. In Set theory of the reals, volume 6 of Israel Mathematical Conference Proceedings, pages 439-561. 1993.
105. [106] S. Shelah. Proper Forcing. Lecture Notes in Mathematics 940. Springer, 1982.
106. [107] S. Shelah and J. Steprans. PFA implies all automorphisms are trivial. Proceedings of the American Mathematical Society, 104:1220-1225, 1988.
107. [108] S. Shelah and J. Steprans. Somewhere trivial automorphisms. Journal of the London Mathematical Society, 49:569-580, 1994.
108. [109] S. Shelah and J. Steprans. Martin's axiom is consistent with the existence of nowhere trivial automorphisms. preprint, Rutgers University, 2000.
109. [110] J.R. Shoenfield. The problem of predicativity. In Y. Bar-Hillel et al., editors, Essays on the Foundations of Mathematics, pages 132-142. The Magens Press, Jerusalem, 1961.
110. [111] P. Simon. Applications of independent linked families. In Topology, theory and applications (Eger, 1983), pages 561-580. North-Holland, Amsterdam, 1985.
111. [112] S. Solecki. Analytic ideals. The Bulletin of Symbolic Logic, 2:339-348, 1996.
112. [113] S. Solecki. an untitled handwritten note. University of Indiana, Bloomington, 1997.
113. [114] S. Solecki. Analytic ideals and their applications. Annals of Pure and Applied Logic, 99:51-72, 1999.
114. δ subsets of analytic ideals of compact sets. handwritten note, University of Indiana, Bloomington, 1999.
115. [116] S. Solecki and S. Todorcevic. Borel bases for analytic ideals. preprint, University of Toronto, 1998.
116. [117] R. Solovay. A model of set theory in which every set of reals is Lebesgue measurable. Annals of Mathematics, 92:1-56, 1970.
117. [118] R. Solovay. Discontinuous homomorphisms of Banach algebras. preprint, UC Berkely, 1976.
118. [119] J. Steprans. The size of the autohomeomorphism group of the Čech-Stone compactification of the integers. preprint, York University, 2000.
119. [120] E. Szemeredi. On sets of integers containing no k elements in arithmetic progression. Acta Arithmetica, 27:199-245, 1975.
120. [121] M. Talagrand. Compacts de fonctions mesurables et filters nonmesurables. Studia Mathematica, 67:13-43, 1980.
121. [122] M. Talagrand. A simple example of a pathological submeasure. Mathematische Annalen, 252:97-102, 1980.
122. [123] D. E. Talayco. Applications of Cohomology to Set Theory I: Hausdorff gaps. Annals of Pure and Applied Logic, 71:69-106, 1995.
123. [124] S. Todorcevic. Directed sets and cofinal types. Transactions of the American Mathematical Society, 290:711-723, 1985.
124. [125] S. Todorcevic. Partition Problems in Topology, volume 84 of Contemporary mathematics. American Mathematical Society, Providence, Rhode Island, 1989.
125. [126] S. Todorcevic. Analytic gaps. Fundamenta Mathematicae, 150:55-67, 1996.
126. [127] S. Todorcevic. Comparing the continuum with the first two uncountable cardinals. In M.L. Dalla Chiara et al., editors, Proceedings of the 10th International Congress of Logic, Methodology and Philosophy of Science, Florence 1995, pages 145-155. Kluwer Academic Publishers, 1997.
127. [128] S. Todorcevic. Definable ideals and gaps in their quotients. In C.A. DiPrisco et al., editors, Set Theory: Techniques and Applications, pages 213-226. Kluwer Academic Press, 1997.
128. σ-sets. J. Math. Soc. Japan, 50:831-836, 1998.
129. [130] S. Todorcevic. Gaps in analytic quotients. Fundamenta Mathematicae, 156:85-97, 1998.
130. [131] S. Todorcevic. Compact sets of Baire class-1 functions. Journal of the American Mathematical Society, 12:1179-1212, 1999.
131. [132] S. Todorcevic. Chain condition methods in topology. Topology and its applications, 101:45-82, 2000.
132. [133] S. Todorcevic. The orthogonal of every analytic P-ideal is countably generated. handwritten note, University of Toronto, July 1996.
133. [134] S. Todorcevic. A dichotomy for P-ideals of countable sets. Fundamenta Mathematicae, to appear.
134. [135] S. Todorcevic and I. Farah. Some Applications of the Method of Forcing. Mathematical Institute, Belgrade and Yenisei, Moscow, 1995.
135. [136] F. Topsøe. Some remarks concerning pathological submeasures. Mathematica Scandinavica, 38:159-166, 1976.
136. 0. Functional Anal. Appl., 8:138-141, 1974. translated from the Russian.
137. [138] J.W. Tukey. Convergence and uniformity in topology. Princeton University Press, 1940.
138. [139] S.M. Ulam. Problems in modern mathematics. John Wiley & Sons, 1964.
139. [140] S.M. Ulam and D. Mauldin. Mathematical problems and games. Advances in Applied Mathematics, 8:281-344, 1987.
140. [141] B. Velickovic. Definable automorphisms of P(ω)/Fin. Proceedings of the American Mathematical Society, 96:130-135, 1986.
141. [142] B. Velickovic. OCA and automorphisms of P(ω)/Fin. Topology and its Applications, 49:1-12, 1992.
142. [143] P. Štěpánek and M. Rubin. Homogeneous Boolean algebras. In D. Monk and R. Bonnett, editors, Handbook of Boolean algebras, volume 2, pages 679-715. North-Holland Amsterdam, 1989.
143. [144] W. Weiss. Partitioning topological spaces. In J. Nešetřil and V. Rödl, editors, Mathematics of Ramsey Theory, volume 5 of Algorithms and Combinatorics, pages 154-171. Springer-Verlag, Berlin, 1990.
144. 1-absoluteness. Note of May 1985.
145. [146] W.H. Woodin. The Axiom of Determinacy, forcing axioms and the nonstationary ideal, volume 1 of de Gruyter Series in Logic and Its Applications. de Gruyter, 1999.