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Boolean powers of groups

Published online by Cambridge University Press:  24 October 2008

A. B. Apps
A. B. Apps
Affiliation:
Christ's College, Cambridge
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If M is any algebraic structure, and R is any Boolean ring, then a structure called the (bounded) Boolean power of M by R, denoted MR, can be defined. This construction, which is also called a bounded Boolean extension, is a sort of generalized direct power, and was introduced by Foster in the 1950's (as a refinement of his previous notion of a Boolean extension). In this paper we shall study isomorphism types and automorphisms of Boolean powers of groups, and obtain information about their characteristic subgroups: we shall be chiefly concerned with Boolean powers of finite groups.

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Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 91 , Issue 3 , May 1982 , pp. 375 - 396
Copyright
Copyright © Cambridge Philosophical Society 1982

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References

REFERENCES

(1)Apps, A. B. On the structure of ℵo-categorical groups, to be submitted to Journal of Algebra.Google Scholar
(2)Balbes, R. and Dwinger, P.Distributive lattices (University of Missouri Press, 1974).Google Scholar
(3)Barwise, J. Back and forth through infinitary logic. In Studies in Model Theory, ed. Morley, M. D., pp. 5–34. (MAA Studies, vol. 8, 1973.)Google Scholar
(4)Burris, S.Boolean powers. Algebra Universalis 5 (1975), 341–360.CrossRefGoogle Scholar
(5)Foster, A. L.Functional completeness in the small. Algebraic structure theorems and identities. Math. Ann. 143 (1961), 29–58.CrossRefGoogle Scholar
(6)Fuchs, L.Abelian groups (Pergamon Press, New York, 1960).Google Scholar
(7)Hall, P.On the embedding of a group in a join of given groups. J. Austral. Math. Soc. 17 (1974), 434–495.CrossRefGoogle Scholar
(8)Hickin, K. and Plotkin, J. M.Boolean powers: direct decomposition and isomorphism types. Trans. Amer. Math. Soc. 265 (1981), 607–621.CrossRefGoogle Scholar
(9)JóNsson, B.On isomorphism types of groups and other algebraic systems. Math. Scand. 5 (1957), 224–229.CrossRefGoogle Scholar
(10)Ketonen, J.The structure of countable Boolean algebras. Ann. of Math. 108 (1978), 41–89.CrossRefGoogle Scholar
(11)Neumann, B. H. and Yamamuro, S.Boolean powers of simple groups. J. Austral. Math. Soc. 5 (1965), 315–324.CrossRefGoogle Scholar
(12)Phillips, R. E.Countably recognizable classes of groups. Rocky Mountain J. Math. 1 (1971), 489–497.CrossRefGoogle Scholar
(13)Scott, W. R.Group theory (Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1964).Google Scholar
(14)Sikorski, R.Boolean algebras (Springer-Verlag, Berlin-Gottingen-Heidelberg, 1960).Google Scholar
(15)Stone, M. H.Topological representations of distributive lattices and Brouwerian logics. Časopis Pěst. Mat. 67 (1937), 1–25.Google Scholar
(16)Waszkiewicz, J. and Weglorz, B.On ω0-categoricity of powers. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys. 17 (1969), 195–199.Google Scholar
(17)Wilson, J. S.Groups with many characteristically simple subgroups. Proc. Cambridge Philos. Soc. 86 (1979), 193–197.CrossRefGoogle Scholar