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Boolean simple groups and boolean simple rings

  • Gaisi Takeuti (a1)

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Let be a complete Boolean algebra and G a finite simple group in the Scott-Solovay -valued model V() of set theory. If we observe G outside V(), then we get a new group which is denoted by Ĝ. In general, Ĝ is not finite nor simple. Nevertheless Ĝ satisfies every property satisfied by a finite simple group with some translation. In this way, we can get a class of groups for which we can use a well-developed theory of the finite simple groups. We call Ĝ Boolean simple if G is simple in some V(). In the same way we define Boolean simple rings. The main purpose of this paper is a study of structures of Boolean simple groups and Boolean simple rings. As for Boolean simple rings, K. Eda previously constructed Boolean completion of rings with a certain condition. His construction is useful for our purpose.

The present work is a part of a series of systematic applications of Boolean valued method. The reader who is interested in this subject should consult with papers by Eda, Nishimura, Ozawa, and the author in the list of references.

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[1]Eda, K. and Hibino, K., On Boolean powers of the group Z and (ω, ω)-weak distributivity, Journal of the Mathematical Society of Japan, vol. 36 (1984), pp. 619628.
[2]Eda, K., A Boolean power and a direct product of abelian groups, Tsukuba Journal of Mathematics, vol. 6 (1982), pp. 187194.
[3]Eda, K., On a Boolean power of a torsionfree abelian group, Journal of Algebra, vol. 82 (1983), pp. 8394.
[4]Lang, S., Algebra, Addison-Wesley, Reading, Massachusetts, 1971.
[5]Kamo, S., On the slender property of certain Boolean algebras, Journal of the Mathematical Society of Japan, vol. 38 (1986), pp. 493500.
[6]Mansfield, R., The theory of Boolean ultrapowers, Annals of Mathematical Logic, vol. 2 (1971), pp. 297323.
[7]Nishimura, H., An approach to the dimension theory of continuous geometry from the standpoint of Boolean valued analysis, Publications of the Research Institute for the Mathematical Sciences, vol. 21 (1985), pp. 181190.
[8]Ozawa, M., Boolean valued interpretation of Hilbert space theory, Journal of the Mathematical Society of Japan, vol. 35 (1983), pp. 609627.
[9]Ozawa, M., Boolean valued analysis and type I AW*-algebras, Proceedings of the Japan Academy, vol. 59 (1983), pp. 368371.
[10]Ozawa, M., A classification of type I AW*-algebras and Boolean valued analysis, Journal of the Mathematical Society of Japan, vol. 36 (1984), pp. 589608.
[11]Ozawa, M., A transfer principle from von Neumann algebras to AW*-algebras, Journal of the London Mathematical Society, ser. 2, vol. 32 (1985), pp. 141148.
[12]Takeuti, G., Two applications of logic to mathematics, Publications of the Mathematical Society of Japan, no. 13, Iwanami Shoten, Tokyo, and Princeton University Press, Princeton, New Jersey, 1978.
[13]Takeuti, G., A transfer principle in harmonic analysis, this Journal, vol. 44 (1979), pp. 417440.
[14]Takeuti, G., Boolean valued analysis, Applications of sheaves (Proceedings, Durham, England, 1977; Fourman, M. P.et al., editors), Lecture Notes in Mathematics, vol. 753, Springer-Verlag, Berlin, 1979, pp. 714731.
[15]Takeuti, G., Boolean completion and m-convergence, Categorical aspects of topology and analysis (Proceedings, Ottawa, 1980; Banaschewski, B., editor), Lecture Notes in Mathematics, vol. 915, Springer-Verlag, Berlin, 1982, pp. 333350.
[16]Takeuti, G., Von Neumann algebras and Boolean valued analysis, Journal of the Mathematical Society of Japan, vol. 35 (1983), pp. 121.
[17]Takeuti, G., C*-algebras and Boolean valued analysis, Japanese Journal of Mathematics, vol. 9 (1983), pp. 207245.

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