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

Journal of Symbolic Logic 50 (1):59-71 (1985)

Abstract

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 α |geq 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 $\kappa \geq|\alpha|$ , and ones with a single generator

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References found in this work

Cylindric Algebras.Leon Henkin & Alfred Tarski - 1967 - Journal of Symbolic Logic 32 (3):415-416.
Classification Theory and the Number of Nonisomorphic Models.S. Shelah - 1982 - Journal of Symbolic Logic 47 (3):694-696.

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