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Completely Additive Liftings

  • Ilijas Farah (a1) (a2)

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The purpose of this communication is to survey a theory of liftings, as developed in author's thesis ([8]). The first result in this area was Shelah's construction of a model of set theory in which every automorphism of P(ℕ)/ Fin, where Fin is the ideal of finite sets, is trivial, or inother words, it is induced by a function mapping integers into integers ([33]). (It is a classical result of W. Rudin [31] that under the Continuum Hypothesis there are automorphisms other than trivial ones.) Soon afterwards, Velickovic ([47]), was able to extract from Shelah's argument the fact that every automorphism of P(ℕ)/ Fin with a Baire-measurable lifting has to be trivial. This, for instance, implies that in Solovay's model ([36]) all automorphisms are trivial. Later on, an axiomatic approach was adopted and Shelah's conclusion was drawn first from the Proper Forcing Axiom (PFA) ([34]) and then from the milder Open Coloring Axiom (OCA) and Martin's Axiom (MA) ([48], see §5 for definitions). Both shifts from the quotient P(ℕ)/ Fin to quotients over more general ideals P(ℕ)/I and from automorphisms to arbitrary ho-momorphisms were made by Just in a series of papers ([14]-[17]), motivated by some problems in algebra ([7, pp. 38–39], [43, I.12.11], [45, Q48]) and topology ([46, p. 537]).

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[1] Boppana, R. B. and Sipser, M., The complexity of finite functions, Handbook of theoretical computer science (van Leeuwen, J., editor), Elsevier, 1990, pp. 757804.
[2] Christensen, J. P. R., Some results with relation to the control measure problem, Vector measures and applications, Springer Lecture Notes in Mathematics, no. 644, Springer-Verlag, Berlin, 1978, pp. 125158.
[3] Comfort, W. W. and Negrepontis, S., Theory of ultrafilters, Springer-Verlag, Berlin, 1974.
[4] Dow, A. and Hart, K. P., ω* has (almost) no continuous images, preprint, 1997.
[5] Dow, A., Simon, P., and Vaughan, J. E., Strong homology and the proper forcing axiom, Proceedings of the American Mathematical Society, vol. 106 (1989), no. 3, pp. 821828.
[6] Engelking, R., General topology, Heldermann, Berlin, 1989.
[7] Erdös, P., My Scottish book “problems”, The Scottish book (Mauldin, D., editor), Birkhäuser, Boston, 1981, pp. 3544.
[8] Farah, I., Analytic ideals and their quotients, Ph.D. thesis , University of Toronto, 1997.
[9] Farah, I., Analytic quotients, submitted, 1997.
[10] Farah, I., Approximate homomorphisms, submitted, 1997.
[11] Hjorth, G. and Kechris, A., New dichotomies for Borel equivalence relations, this Bulletin, vol. 3 (1997), pp. 329346.
[12] Jalali-Naini, S.-A., The monotone subsets of Cantor space, filters and descriptive set theory, Doctoral dissertation, Oxford, 1976.
[13] Just, W., Nowhere dense P-subsets of ω*, Proceedings of the American Mathematical Society, vol. 106 (1989), pp. 11451146.
[14] Just, W., The space (ω*)n+1 is not always a continuous image of (ω*)n , Fundamenta Mathematicae, vol. 132 (1989), pp. 5972.
[15] Just, W., Repercussions on a problem of Erdős and Ulam about density ideals, Canadian Journal of Mathematics, vol. 42 (1990), pp. 902914.
[16] Just, W., A modification of Shelah's oracle chain condition with applications, Transactions of the American Mathematical Society, vol. 329 (1992), pp. 325341.
[17] Just, W., A weak version of AT from OCA, Mathematical Sciences Research Institute Publications (1992), no. 26, pp. 281291.
[18] Just, W. and Krawczyk, A., On certain Boolean algebras P(ω)/I, Transactions of the American Mathematical Society, vol. 285 (1984), pp. 411429.
[19] Kalton, N. J., The Maharam problem, Séminaire Initiation à ĺ Analyse, vol. 18 (19881989), pp. 113.
[20] Kalton, N. J. and Roberts, J. W., Uniformly exhaustive submeasures andnearly additive set functions, Transactions of the American Mathematical Society, vol. 278 (1983), pp. 803816.
[21] Kechris, A. S., Rigidity properties of Borel ideals on the integers, Topology and its Applications, to appear.
[22] Kechris, A. S. and Louveau, A., The structure of hypersmooth Borel equivalence relations, Journal of the American Mathematical Society, vol. 10, no. 1, pp. 215242.
[23] Kunen, K., 〈κ, λ*#x232A;-gaps under MA, preprint, 1976.
[24] Louveau, A., Progres recents sur le probleme de Maharam d'apres N. J. Kalton et J. W. Roberts, Séminaire Initiation à ĺ Analyse, vol. 20 (19831984), pp. 18.
[25] Louveau, A. and Velickovic, B., A note on Borel equivalence relations, Proceedings of the American Mathematical Society, vol. 120 (1994), pp. 255259.
[26] Mardesic, S. and Prasolov, A., Strong homology is not additive, Transactions of the American Mathematical Society, vol. 307 (1988), pp. 725744.
[27] Mathias, A. R. D., A remark on rare filters, Infinite and finite sets (Hajnal, A. et al., editors), Colloquia Mathematica Societatis János Bolyai, no. 10, North-Holland, 1975.
[28] Maztjr, K., Fσ-ideals and ω1ω1*-gaps in the Boolean algebra P(ω)/I, Fundamenta Mathematicae, vol. 138 (1991), pp. 103111.
[29] Maztjr, K., A modification of Louveau and Velickovic construction for Fσ -ideals, preprint, 1996.
[30] Parovičenko, I.I., A universal bicompact of weight ℵ1 , Soviet Mathematics Doklady, vol. 4 (1963), pp. 592592.
[31] Rudin, W., Homogeneity problems in the theory of Čech compactifications, Duke Mathematics Journal, vol. 23 (1956), pp. 409419.
[32] Scheepers, M., Gaps in ωω , Israel Mathematical Conference Proceedings, vol. 6 (1993), pp. 439561.
[33] Shelah, S., Proper forcing, Springer Lecture Notes in Mathematics, no. 940, Springer-Verlag, Berlin, 1982.
[34] Shelah, S. and Steprans, J., PFA implies all automorphisms are trivial, Proceedings of the American Mathematical Society, vol. 104 (1988), pp. 12201225.
[35] Solecki, S., Analytic ideals, this Bulletin, vol. 2 (1996), pp. 339348.
[36] Solovay, R., A model of set theory in which every set of reals is Lebesgue measurable, Annals of Mathematics, vol. 92 (1970), pp. 156.
[37] Talagrand, M., Compacts de fonctions mesurables et filters nonmesurables, Studia Mathematica, vol. 67 (1980), pp. 1343.
[38] Todorcevic, S., The first derived limit and compactly Fσ-sets, Journal of the Mathematical Society of Japan, to appear.
[39] Todorcevic, S., Gaps in analytic quotients, Fundamenta Mathematicae, to appear.
[40] Todorcevic, S., Partition problems in topology, American Mathematical Society, Providence, Rhode Island, 1989.
[41] Todorcevic, S., Analytic gaps, Fundamenta Mathematicae, vol. 150 (1996), pp. 5566.
[42] Todorcevic, S., Definable ideals and gaps in their quotients, preprint, 1997.
[43] Ulam, S. M., Problems in modern mathematics, John Wiley & Sons, 1964.
[44] Ulam, S. M. and Mauldin, D., Mathematical problems and games, Advances in Applied Mathematics, vol. 8 (1987), pp. 281344.
[45] van Douwen, E., Monk, D., and Rubin, M., Some questions about Boolean algebras, Algebra Universalis, vol. 11 (1980), pp. 220243.
[46] van Mill, J., An introduction to βω, Handbook of set-theoretic topology (Kunen, K. and Vaughan, J., editors), North-Holland, 1984, pp. 503568.
[47] Velickovic, B., Definable automorphisms of P(ω)/ Fin, Proceedings of the American Mathematical Society, vol. 96 (1986), pp. 130135.
[48] Velickovic, B., OCA and automorphisms of P(ω)/ Fin, Topology and its Applications, vol. 49 (1992), pp. 112.

Completely Additive Liftings

  • Ilijas Farah (a1) (a2)

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