Journal of Symbolic Logic

The Number of One-Generated Cylindric Set Algebras of Dimension Greater than Two

Jean A. Larson

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S. Ulam asked about the number of nonisomorphic projective algebras with $k$ 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 $\alpha |geq 2$, the number of isomorphism classes of regular locally finite cylindric set algebras of dimension $\alpha$ of the following two kinds are computed: ones of power $\kappa$ for infinite $\kappa \geq |\alpha|$, and ones with a single generator.

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J. Symbolic Logic, Volume 50, Issue 1 (1985), 59-71.

First available in Project Euclid: 6 July 2007

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Larson, Jean A. The Number of One-Generated Cylindric Set Algebras of Dimension Greater than Two. J. Symbolic Logic 50 (1985), no. 1, 59--71.

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