Hostname: page-component-8448b6f56d-gtxcr Total loading time: 0 Render date: 2024-04-20T03:40:09.906Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[0] Andréka, H. and Németi, I., On the number of generators of cylindric algebras, preprint, Institute of Mathematics, Hungarian Academy of Sciences, Budapest, 1979.Google Scholar
[1] Bergman, G., The rank of a two-dimensional cylindric set algebra, Universal algebra (Esztergom, 1977), Colloquia Mathematica Societatis János Bolyai, vol. 29, North-Holland, Amsterdam, 1982, pp. 95–105.Google Scholar
[2] Chin, L. H. and Tarski, A., Remarks on projective algebras (abstract), Bulletin of the American Mathematical Society, vol. 54 (1948), pp. 80–81.Google Scholar
[3] Erdös, P., Faber, V. and Larson, J., Sets of natural numbers of positive density and cylindric set algebras of dimension 2, Algebra Universalis, vol. 12 (1981), pp. 81–92.Google Scholar
[4] Everett, C. J. and Ulam, S. M., Projective algebra. I, American Journal of Mathematics, vol. 68 (1946), pp. 77–88.Google Scholar
[5] Henkin, L., Monk, J. D. and Tarski, A., Cylindric algebras. Part I, North-Holland, Amsterdam, 1971.Google Scholar
[6] Henkin, L. et al., Cylindric set algebras, Lecture Notes in Mathematics, vol. 883, Springer-Verlag, Berlin, 1981.CrossRefGoogle Scholar
[7] Larson, J. A., The number of one-generated finite dimensional cylindric set algebras (abstract 78T-A51), Notices of the American Mathematical Society, vol. 25 (1978), p. A–226.Google Scholar
[8] Larson, J. A., The number of one-generated diagonal-free cylindric set algebras of finite dimension greater than two, Algebra Universalis, vol. 16 (1983), pp. 1–16.Google Scholar
[9] Larson, J. A., The number of finitely generated infinite cylindric algebras of dimension two, Algebra Universalis (to appear).Google Scholar
[10] Shelah, S., Classification theory and the number of nonisomorphic models, North-Holland, Amsterdam, 1978.Google Scholar
[11] Ulam, S. M., A collection of mathematical problems, Wiley/Interscience, New York, 1960.Google Scholar
[12] Comer, S., Galois theory for cylindric set algebras and its applications, Transactions of the American Mathematical Society (to appear).Google Scholar
[13] Maddux, R., The equational theory of CA3 is undecidable, this Journal, vol. 45 (1980), pp. 311–316.Google Scholar