Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-28T16:36:53.457Z Has data issue: false hasContentIssue false

The lattice of varieties of representable relation algebras

Published online by Cambridge University Press:  12 March 2014

Steven Givant
Affiliation:
Department of Mathematics, Mills College, Oakland, California 94613, E-mail: [email protected]
István Németi
Affiliation:
Mathematical Institute of the Hungarian Academy of Sciences, P.O. Box 127, Budapest V, H-1364Hungary, E-mail: [email protected]

Abstract

We shall show that certain natural and interesting intervals in the lattice of varieties of representable relation algebras embed the lattice of all subsets of the natural numbers, and therefore must have a very complicated lattice-theoretic structure.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1994

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

[1990] Andréka, H., Givant, S., and Németi, I., Essentially infinite varieties of relation algebras, Abstracts of papers presented to the American Mathematical Society, vol. 11 (1990), p. 230.Google Scholar
[1993] Blok, W. and Pigozzi, D., On the structure of varieties with equationally definable principal congruences. Part IV, Algebra Universalis (to appear).Google Scholar
[1983] Comer, S., A remark on chromatic polygroups, Congressus Numerantium, vol. 38 (1983), pp. 8595.Google Scholar
[1985] Droste, M., Structure of partially ordered sets with transitive automorphism groups, Memoirs of the American Mathematical Society, vol. 57, no. 334, American Mathematical Society, Providence, Rhode Island, 1985.Google Scholar
[1991] Düntsch, I., Small integral relation algebras generated by a partial order, Periodica Mathematica Hungarica, vol. 23 (1991), pp. 129138.Google Scholar
[1982] Jónsson, B., Varieties of relation algebras, Algebra Universalis, vol. 15 (1982), pp. 273298.CrossRefGoogle Scholar
[1988] Jónsson, B., Relation algebras and Schröder categories, Discrete Mathematics, vol. 70 (1988), pp. 2745.CrossRefGoogle Scholar
[1952] Jónsson, B. and Tarski, A., Boolean algebras with operators, Part II, American Journal of Mathematics, vol. 74 (1952), pp. 127162.Google Scholar
[1955] Ëoś, J., Quelques remarques, théorèmes et problèmes sur les classes définissables d'algèbres, Mathematical interpretation of formal systems, North-Holland, Amsterdam, 1955, pp. 98113.Google Scholar
[1950] Lyndon, R., The representation of relational algebras, Annals of Mathematics, vol. 51 (1950), pp. 707729.Google Scholar
[1978] Maddux, R. D., Topics in relation algebras, Doctoral dissertation , University of California at Berkeley, Berkeley, California, 1978.Google Scholar
[1966] McKenzie, R. N., The representation of relation algebras, Doctoral dissertation , University of Colorado at Boulder, Boulder, Colorado, 1966.Google Scholar
[1964] Monk, J. D., On representable relation algebras, Michigan Mathematics Journal, vol. 11 (1964), pp. 207210.Google Scholar
[1992] Németi, I., Algebraizations of quantifier logics, an introductory overview. Version 10.2, Mathematical Institute of the Hungarian Academy of Science, Budapest, preprint (1992); A shorter version (without proofs) appeared in Studia Logica , vol. 50 (1991), pp. 485–570.Google Scholar
[1967] Neumann, H., Varieties of groups, Ergebnisse der Mathematik und Ihrer Grenzgebiete, vol. 37 (1967), Springer-Verlag, Berlin, Heidelberg, New York.Google Scholar
[1970] Ol'šanskiĭ, A. Ju., On the problem of a finite basis of identities in groups, Mathematics of the USSR-lzvestija, vol. 4 (1970), pp. 381389.Google Scholar
[1941] Tarski, A., On the calculus of relations, this Journal, vol. 6 (1941), pp. 7389.Google Scholar
[1954] Tarski, A., Contributions to the theory of models. I, Koninklijke Nederlandse Academie van Wetenschappen, Proceedings, Series A vol. 57 ( Indagationes Mathematical vol. 16) (1954), pp. 572581.Google Scholar
[1955] Tarski, A., Contributions to the theory of models. III, Koninklijke Nederlandse Academie van Wetenschappen, Proceedings, Series A, vol. 58 ( Indagationes Mathematicae vol. 17) (1955), pp. 5664.Google Scholar
[1956] Tarski, A., Equationally complete rings and relation algebras, Koninklijke Nederlandse Academie van Wetenschappen, Proceedings, Series A vol. 59 ( Indagationes Mathematicae vol. 18) (1956), pp. 3946.Google Scholar
[1970] Vaughn-Lee, M. R., Uncountably many varieties of groups, Bulletin of the London Mathematical Society, vol. 2 (1970), pp. 280286.Google Scholar
[1978] Werner, H., Discriminator algebras, Studien zur Algebra und Ihre Anwendungen, vol. 6, Akademie-Verlag, Berlin, 1978.Google Scholar