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Universal classes of simple relation algebras

Published online by Cambridge University Press:  12 March 2014

Steven Givant
Steven Givant*
Affiliation:
Department of Mathematics and Computer Science, Mills College, Oakland, CA 94613, USA E-mail: [email protected]
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Extract

Tarski [19] proved the important theorem that the class of representable relation algebras is equationally axiomatizable. One of the key steps in his proof is showing that the class of (isomorphs of) simple set relation algebras—that is, algebras of binary relations with a unit of the form U × U for some non-empty set U —is universal, i.e., is axiomatizable by a set of universal sentences. In the same paper Tarski observed that the class of (isomorphs of) relation algebras constructed from groups (so-called group relation algebras) is also universal.

We shall abstract the essential ingredients of Tarski's method (in Corollary 2.4), and then combine them with some observations about atom structures, to establish (in Theorem 2.6) a rather general method for showing that certain classes of simple relation algebras—and, more generally, certain classes of simple algebras in a discriminator variety V—are universal, and consequently that the collections of (isomorphs of) subdirect products of algebras in such classes form subvarieties of V. As applications of the method we show that two well-known classes of simple relation algebras, those constructed from projective geometries (sometimes called Lyndon algebras) and those constructed from modular lattices with a zero (sometimes called Maddux algebras), are universal. In the process we prove that these two classes consist precisely of all (isomorphs of) complex algebras over the respective geometries and modular lattices, provided that we choose the primitive notions of the latter structures in an appropriate fashion. We also derive Tarski's theorems and a related theorem of the author as easy corollaries of Theorem 2.6.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 64 , Issue 2 , June 1999 , pp. 575 - 589
Copyright
Copyright © Association for Symbolic Logic 1999

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References

REFERENCES

[1] Andréka, H., Givant, S., and Németi, I., Decision problems for equational theories of relation algebras, Memoirs of the American Mathematical Society, vol. 604, American Mathematical Society, Providence, RI, 1997.Google Scholar
[2] Andréka, H., Jó, B., and Németi, I., Free algebras in discriminator varieties, Algebra Universalis, vol. 28 (1991), pp. 401–447.CrossRefGoogle Scholar
[3] Blackburn, P., de Rijke, M., and Venema, Y., The algebra of modal logic, Technical report IR-370, Department of Computer Science, Free University of Amsterdam, Amsterdam, 1994.Google Scholar
[4] Fine, K., Some connections between elementary and modal logic, Proceedings of the third Scandinavian logic symposium (Kanger, S., editor), North-Holland Publishing Company, Amsterdam, 1975, pp. 15–31.Google Scholar
[5] Goldblatt, R., Varieties of complex algebras, Annals of Pure and Applied Logic, vol. 44 (1989), pp. 173–242.CrossRefGoogle Scholar
[6] Henkin, L., Monk, J. D., and Tarski, A., Cylindric algebras, Part II, Studies in Logic and the Foundations of Mathematics, vol. 115, North-Holland Publishing Company, Amsterdam, 1985.Google 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., Varieties of relation algebras, Algebra Universalis, vol. 15 (1982), pp. 273–298.CrossRefGoogle Scholar
[9] Jónsson, B., The theory of binary relations, Algebraic logic (Andréka, H., Monk, J. D., and Németi, I., editors), North-Holland Publishing Company, Amsterdam, 1991, pp. 245–292.Google Scholar
[10] JóNsson, B. and Tarski, A., Boolean algebras with operators, Part I, American Journal of Mathematics, vol. 73 (1951), pp. 891–939.CrossRefGoogle Scholar
[11] Jónsson, B. and Tarski, A., Boolean algebras with operators, Part II, American Journal of Mathematics vol. 74 (1952), pp. 127–162.Google Scholar
[12] Łoś, J., Quelques remarques, théorèmes et problèlmes sur les classes définissables d'algèbres, Mathematical interpretation of formal systems, Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Company, Amsterdam, 1955, pp. 98–113.Google Scholar
[13] Lyndon, R. C., Relation algebras and projective geometries, Michigan Mathematical Journal, vol. 8 (1961), pp. 21–28.CrossRefGoogle Scholar
[14] Maddux, R., Embedding modular lattices into relation algebras, Algebra Universalis, vol. 12 (1981), pp. 242–246.CrossRefGoogle Scholar
[15] McKenzie, R., The representation of relation algebras. Doctoral dissertation , University of Colorado at Boulder, Boulder, CO, 1966.Google Scholar
[16] Monk, J. D., Connections between combinatorial theory and algebraic logic, Studies in algebraic logic (Daigneault, A., editor), Studies in Mathematics, vol. 9, Mathematical Association of America, Washington D. C., 1974, pp. 58–91.Google Scholar
[17] Németi, I., Algebraizations of quantifier logics, an introductory overview, Preprint 50/1993, Mathematical Institute of the Hungarian Academy of Sciences, Budapest, 1993.Google Scholar
[18] Schwabhäuser, W., Szmielew, W., and Tarski, A., Metamathematische Methoden in der Geometric, Hochschultext, Springer-Verlag, Berlin, 1983.CrossRefGoogle Scholar
[19] Tarski, A., Contributions to the theory of models, III, Koninklijke Nederlandse Akademie van Wetenschappen, Proceedings, Series A, Mathematical Sciences, vol. 58 (1955), pp. 56–64, Indagationes Mathematicae , vol. 17.Google Scholar
[20] Tarski, A. and Givant, S., A formalization of set theory without variables, Colloquium Publications, vol. 41, American Mathematical Society, Providence, RI, 1987.Google Scholar
[21] van Benthem, J., Some kinds of modal completeness, Stadia Logica, vol. 39 (1980), pp. 125–141.Google Scholar
[22] Venema, Y., Many-dimensional modal logic, Doctoral dissertation , University of Amsterdam, Amsterdam, 1992.Google Scholar
[23] Werner, H., Discriminator algebras, Studien zur Algebra und ihre Anwendungen, vol. 6, Akademie Verlag, Berlin, 1978.Google Scholar