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Relation algebras of every dimension

Published online by Cambridge University Press:  12 March 2014

Roger D. Maddux
Roger D. Maddux*
Affiliation:
Department of Mathematics, Iowa State University, Ames, Iowa 50011, E-mail: [email protected]
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Abstract

Conjecture (1) of [Ma83] is confirmed here by the following result: if 3 ≤ α < ω, then there is a finite relation algebra of dimension α, which is not a relation algebra of dimension α + 1. A logical consequence of this theorem is that for every finite α ≥ 3 there is a formula of the form S ⊆ T (asserting that one binary relation is included in another), which is provable with α + 1 variables, but not provable with only α variables (using a special sequent calculus designed for deducing properties of binary relations).

Keywords

Logiccomplexity of proofsprovability with finitely many variablesrelation algebrasrelational basisdimension
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 57 , Issue 4 , December 1992 , pp. 1213 - 1229
Copyright
Copyright © Association for Symbolic Logic 1992

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References

REFERENCES

[HMT85]Henkin, L., Monk, J. D., and Tarski, A., Cylindric Algebras, Part II, North-Holland, Amsterdam, 1985.Google Scholar
[JT51]Jónsson, Bjarni and Tarski, Alfred, Boolean algebras with operators, Part I, American Journal of Mathematics, vol. 73 (1951), pp. 891–939.CrossRefGoogle Scholar
[JT52]Jónsson, Bjarni and Tarski, Alfred, Boolean algebras with operators, Part II, American Journal of Mathematics, vol. 74 (1952), pp. 127–162.CrossRefGoogle Scholar
[Ly50]Lyndon, Roger C., The representation of relational algebras, Annals of Mathematics, vol. 51 (1950), pp. 707–729.CrossRefGoogle Scholar
[Ly61]Lyndon, Roger C., Relation algebras and projective geometries, Michigan Mathematical Journal, vol. 8 (1961), pp. 21–28.CrossRefGoogle Scholar
[McK66]McKenzie, Ralph N. W., The representation of relation algebras, Doctoral dissertation, University of Colorado, Boulder, 1966.Google Scholar
[McK70]McKenzie, Ralph N. W., The representation of integral relation algebras, Michigan Mathematical Journal, vol. 17 (1970), pp. 279–287.CrossRefGoogle Scholar
[Ma76]Maddux, Roger D., Some nonrepresentable relation algebras, Notices of the American Mathematical Society, vol. 23 (1976), pp. A-431, A–557.Google Scholar
[Ma78]Maddux, Roger D., Topics in relation algebras, Doctoral dissertation, University of California, Berkeley, 1978.Google Scholar
[Ma82]Maddux, Roger D., Some varieties containing relation algebras, Transactions of the American Mathematical Society, vol. 272 (1982), pp. 501–526.CrossRefGoogle Scholar
[Ma83]Maddux, Roger D., A sequent calculus for relation algebras, Annals of Pure and Applied Logic, vol. 25 (1983), pp. 73–101.CrossRefGoogle Scholar
[Ma89]Maddux, Roger D., Nonfinite axiomatizability results for cylindric and relation algebras, This Journal, vol. 54, pp. 951–974.Google Scholar
[Mo61]Monk, J. Donald, Studies in cylindric algebra, Doctoral dissertation, University of California, Berkeley, 1961.Google Scholar
[Mo64]Monk, J. Donald, On representable relation algebras, Michigan Mathematical Journal, vol. 11 (1964), pp. 207–210.CrossRefGoogle Scholar
[Mo69]Monk, J. Donald, Nonfinitizability of classes of representable cylindric algebras, this Journal, vol. 34 (1969), pp. 331–343.Google Scholar