Boolean simple groups and boolean simple rings

E-ISSN： 1943-5886|53|1|160-173

ISSN： 0022-4812

Source： The Journal of Symbolic Logic, Vol.53, Iss.1, 1988-03, pp. : 160-173

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Abstract

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.