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Boolean simple groups and boolean simple rings

Published online by Cambridge University Press:  12 March 2014

Gaisi Takeuti
Gaisi Takeuti*
Affiliation:
Department of Mathematics, University of Illinois, Urbana, Illinois 61801
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Extract

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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 53 , Issue 1 , March 1988 , pp. 160 - 173
Copyright
Copyright © Association for Symbolic Logic 1988

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References

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