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Weak representations of relation algebras and relational bases

Published online by Cambridge University Press:  12 March 2014

Robin Hirsch ,
Ian Hodkinson  and
Roger D. Maddux
Robin Hirsch
Affiliation:
Department of Computer Science, University College London, Gower Street, London WC1E 6BT, UK, E-mail: [email protected]
Ian Hodkinson
Affiliation:
Department of Computing, Imperial College London, London SW7 2AZ, UK, E-mail: [email protected]
Roger D. Maddux
Affiliation:
Department of Mathematics, Iowa State University, Ames, IA 50011, USA, E-mail: [email protected]
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Abstract

It is known that for all finite n ≥ 5, there are relation algebras with n-dimensional relational bases but no weak representations. We prove that conversely, there are finite weakly representable relation algebras with no n-dimensional relational bases. In symbols: neither of the classes RAn and wRRA contains the other.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 76 , Issue 3 , September 2011 , pp. 870 - 882
Copyright
Copyright © Association for Symbolic Logic 2011

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References

REFERENCES

[1]Andréka, H., Weakly representable but not representable relation algebras, Algebra Universalis, vol. 32 (1994), pp. 31–43.CrossRefGoogle Scholar
[2]Haiman, M., Arguesian lattices which are not linear, Bulletin of the American Mathematical Society, vol. 16 (1987), pp. 121–123.CrossRefGoogle Scholar
[3]Haiman, M., Arguesian lattices which are not type I, Algebra Universalis, vol. 28 (1991), pp. 128–137.CrossRefGoogle Scholar
[4]Hirsch, R. and Hodkinson, I., Relation algebras by games, Studies in Logic and the Foundations of Mathematics, vol. 147, North-Holland, Amsterdam, 2002.Google Scholar
[5]Hodkinson, I. and Mikulás, Sz., Non-finite axiomatizability of reducís of algebras of relations, Algebra Universalis, vol. 43 (2000), pp. 127–156.CrossRefGoogle Scholar
[6]Hodkinson, I. and Venema, Y., Canonical varieties with no canonical axiomatisation, Transactions of the American Mathematical Society, vol. 357 (2005), pp. 4579–4605.CrossRefGoogle Scholar
[7]Jónsson, B., Representation of modular lattices and of relation algebras, Transactions of the American Mathematical Society, vol. 92 (1959), pp. 449–464.CrossRefGoogle Scholar
[8]Jónsson, B., The theory of binary relations, Algebraic logic (Andréka, H., Monk, J. D., and Németi, I., editors), Colloq. Math. Soc. J. Bolyai, vol. 54, North-Holland, Amsterdam, 1991, pp. 245–292.Google Scholar
[9]Lyndon, R., The representation of relational algebras, Annals of Mathematics, vol. 51 (1950), no. 3, pp. 707–729.CrossRefGoogle Scholar
[10]Maddux, R. D., Some varieties containing relation algebras, Transactions of the American Mathematical Society, vol. 272 (1982), no. 2, pp. 501–526.CrossRefGoogle Scholar
[11]Maddux, R. D., A sequent calculus for relation algebras, Annals of Pure and Applied Logic, vol. 25 (1983), pp. 73–101.CrossRefGoogle Scholar
[12]Maddux, R. D., Relation algebras of every dimension, this Journal, vol. 57 (1992), pp. 1213–1229.Google Scholar
[13]Maddux, R. D., Relation algebras, Studies in Logic and the Foundations of Mathematics, vol. 150, Elsevier, Amsterdam, 2006.Google Scholar
[14]Monk, J. D., On representable relation algebras, Michigan Mathematics Journal, vol. 11 (1964), pp. 207–210.CrossRefGoogle Scholar
[15]Pécsi, B., Weakly representable relation algebras form a variety, Algebra Universalis, vol. 60 (2009), pp. 369–380.CrossRefGoogle Scholar
[16]Tarski, A. and Givant, S. R., A formalization of set theory without variables, Colloquium Publications, no. 41, American Mathematical Society, Providence, Rhode Island, 1987.CrossRefGoogle Scholar