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The number of one-generated cylindric set algebras of dimension greater than two

Published online by Cambridge University Press:  12 March 2014

Jean A. Larson
Jean A. Larson*
Affiliation:
University of Florida, Gainesville, Florida 32611
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Abstract

S. Ulam asked about the number of nonisomorphic projective algebras with κ generators. This paper answers his question for projective algebras of finite dimension at least three and shows that there are the maximum possible number, continuum many, of nonisomorphic one-generated structures of finite dimension n, where n is at least three, of the following kinds: projective set algebras, projective algebras, diagonal-free cylindric set algebras, diagonal-free cylindric algebras, cylindric set algebras, and cylindric algebras. The results of this paper extend earlier results to the collection of cylindric set algebras and provide a uniform proof for all the results. Extensions of these results for dimension two are discussed where some modifications on the hypotheses are needed. Furthermore for α ≥ 2, the number of isomorphism classes of regular locally finite cylindric set algebras of dimension α of the following two kinds are computed: ones of power κ for infinite κ ≥ ∣α∣, and ones with a single generator.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 50 , Issue 1 , March 1985 , pp. 59 - 71
Copyright
Copyright © Association for Symbolic Logic 1985

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References

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