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Nonfinitizability of classes of representable cylindric algebras1

Published online by Cambridge University Press:  12 March 2014

J. Donald Monk*
Affiliation:
University of Colorado and University of California, Berkeley

Extract

Cylindric algebras were introduced by Alfred Tarski about 1952 to provide an algebraic analysis of (first-order) predicate logic. With each cylindric algebra one can, in fact, associate a certain, in general infinitary, predicate logic; for locally finite cylindric algebras of infinite dimension the associated predicate logics are finitary. As with Boolean algebras and sentential logic, the algebraic counterpart of completeness is representability. Tarski proved the fundamental result that every locally finite cylindric algebra of infinite dimension is representable.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1969

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Footnotes

1

Research supported in part by NSF Grants GP 7387 and GP 6232-x.

References

[1]Greenwood, R. and Gleason, A., Combinatorial relations and chromatic graphs, Canadian journal of mathematics, vol. 7 (1955), pp. 17.CrossRefGoogle Scholar
[2]Halmos, P., Algebraic logic. IV, Transactions of the American Mathematical Society, vol. 86 (1957), pp. 127.Google Scholar
[3]Henkin, L., La structure algébrique des théories mathématiques, Gauthier-Villars, Paris, 1956.Google Scholar
[4]Henkin, L., Monk, J. D., and Tarski, A., Cylindric algebras, Part I, North-Holland, Amsterdam (to appear).CrossRefGoogle Scholar
[5]Henkin, L. and Tarski, A., Cylindric algebras, Proceedings of symposia in pure mathematics, vol. 2, Amer. Math. Soc., Providence, R.I., 1961, pp. 83113.CrossRefGoogle Scholar
[6]Johnson, J. S., Nonfinitizability of classes of representable polyadic algebras, this Journal, vol. 34 (1969), pp. 344352.Google Scholar
[7]Monk, J. D., On the representation theory for cylindric algebras, Pacific journal of mathematics, vol. 11 (1961), pp. 14471457.CrossRefGoogle Scholar
[8]Monk, J. D.. Model-theoretic methods and results in the theory of cylindric algebras, The theory of models, North-Holland, Amsterdam, 1965, pp. 238250.Google Scholar
[9]Tarski, A., Contributions to the theory of models. III, Indagationes mathematicae, vol. 17 (1955), pp. 5664.CrossRefGoogle Scholar