Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-27T12:15:04.988Z Has data issue: false hasContentIssue false
  • English
  • Français

NONREPRESENTABLE RELATION ALGEBRAS FROM GROUPS

Published online by Cambridge University Press:  13 June 2019

HAJNAL ANDRÉKA ,
ISTVÁN NÉMETI  and
STEVEN GIVANT
HAJNAL ANDRÉKA*
Affiliation:
Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences
ISTVÁN NÉMETI*
Affiliation:
Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences
STEVEN GIVANT*
Affiliation:
Department of Mathematics and Computer Science, Mills College
*
*ALFRÉD RÉNYI INSTITUTE OF MATHEMATICS HUNGARIAN ACADEMY OF SCIENCES BUDAPEST, REÁLTANODA ST. 13-15, H-1053, HUNGARY E-mail: [email protected]
†ALFRÉD RÉNYI INSTITUTE OF MATHEMATICS HUNGARIAN ACADEMY OF SCIENCES BUDAPEST, REÁLTANODA ST. 13-15, H-1053, HUNGARY E-mail: [email protected]
‡DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE MILLS COLLEGE 500 MACARTHUR BOULEVARD OAKLAND, CA 94613 USA E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

A series of nonrepresentable relation algebras is constructed from groups. We use them to prove that there are continuum many subvarieties between the variety of representable relation algebras and the variety of coset relation algebras. We present our main construction in terms of polygroupoids.

Keywords

Primary 03G1520Bxx03GxxSecondary 03E2003C05grouprelation algebraBrandt groupoidpolygroupoidatom structurecomplex algebragroup relation algebranonrepresentable relation algebracoset algebrameasurable relation algebra
Type
Research Article
Information
The Review of Symbolic Logic , Volume 13 , Issue 4 , December 2020 , pp. 861 - 881
Check for updates
Copyright
Copyright © Association for Symbolic Logic 2019 

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

BIBLIOGRAPHY

Andréka, H. & Givant, S. (2018). Coset relation algebras. Algebra Universalis, 79(2), paper 28, 53 pp. Sharit link: https://rdcu.be/L94h.CrossRefGoogle Scholar
Andréka, H. & Givant, S. (2019). A representation theorem for measurable relation algebras with cyclic groups. Transactions of the American Mathematical Society, 371(10), 7175–7198.CrossRefGoogle Scholar
Andréka, H., Givant, S., & Németi, I. (1997). Decision problems for equational theories of relation algebras. Memoirs of the American Mathematical Society, 126(604), xiv+126pp.CrossRefGoogle Scholar
Andréka, H., Maddux, R. D., & Németi, I. (1991). Splitting in relation algebras. Proceedings of the American Mathematical Society, 111(4), 1085–1093.CrossRefGoogle Scholar
van Benthem, J. A. F. K. (1991). Language in Action. Lambdas and Dynamic Logic, Amsterdam: North-Holland. x+349 pp.Google Scholar
Brink, C., Kahl, W., & Schmidt, G. (editors) (1997). Relational Methods in Computer Science. Vienna: Springer Verlag.CrossRefGoogle Scholar
Comer, S. D. (1976). Multi-valued loops, geometries, and algebraic logic. Houston Journal of Mathematics, 2, 373–380.Google Scholar
Comer, S. D. (1983). A new foundation for the theory of relations. Notre Dame Journal of Formal Logic, 24(2), 181–187.CrossRefGoogle Scholar
Comer, S. D. (1984), Combinatorial aspects of relations. Algebra Universalis, 18(1), 77–94.CrossRefGoogle Scholar
Comer, S. D. (1985). The Cayley representation of polygroups. In Corsini, P., editor. Convegno su: Ipergruppi, altre strutture multivoche e loro applicazioni. Udine: Università degli Studi di Udine, pp. 27–34.Google Scholar
Dresher, M. & Ore, O. (1938). Theory of multigroups. American Journal of Mathematics, 60, 705–733.CrossRefGoogle Scholar
Düntsch, I. (2005). Relation algebras and their application in temporal and spatial reasoning. Artificial Intelligence Review, 23, 315–357.CrossRefGoogle Scholar
Givant, S. (2017). Introduction to Relation Algebras. Cham: Springer International Publishing AG. xxxii+572 pp.CrossRefGoogle Scholar
Givant, S. (2017). Advanced Topics in Relation Algebras. Cham: Springer International Publishing AG. xix+605 pp.CrossRefGoogle Scholar
Givant, S. (2018). Relation algebras and groups. Algebra Universalis, 79(2), paper 16, 38pp. Sharit link: https://rdcu.be/LJ2Y.CrossRefGoogle Scholar
Givant, S. & Andréka, H. Notes on measurable relation algebras. Manuscript, 1996–2001. Cca. 500 pages. Part of these notes is published as [1, 2, 15, 18, 19]. Part of the present paper is also based on these notes.Google Scholar
Givant, S. & Andréka, H. (2002). Groups and algebras of relations. The Bulletin of Symbolic Logic, 8, 38–64.CrossRefGoogle Scholar
Givant, S. & Andréka, H. (2018). A representation theorem for measurable relation algebras. Annals of Pure and Applied Logic, 169(11), 1117–1189.CrossRefGoogle Scholar
Givant, S. & Andréka, H. (2018). The variety of coset relation algebras. The Journal of Symbolic Logic, 83(4), 1595–1609.CrossRefGoogle Scholar
Hirsch, R. & Hodkinson, I. (2001). Representability is not decidable for finite relation algebras. Transactions of the American Mathematical Society, 353, 1403–1425.CrossRefGoogle Scholar
Hirsch, R. & Hodkinson, I. (2002). Relation Algebras by Games. Amsterdam: North-Holland. 712 pp.Google Scholar
Jónsson, B. (1959). Representation of modular lattices and of relation algebras. Transactions of the American Mathematical Society, 92, 449–464.CrossRefGoogle Scholar
Jónsson, B. (1982). Varieties of relation algebras. Algebra Universalis, 15, 273–298.CrossRefGoogle Scholar
Jónsson, B. (1991). The theory of binary relations. In Andréka, H., Monk, J. D., and Németi, I., editors. Algebraic Logic. Amsterdam: North-Holland, pp. 245–292.Google Scholar
Jónsson, B. & Tarski, A. (1952). Boolean algebras with operators. Part II. American Journal of Mathematics, 74, 127–162.CrossRefGoogle Scholar
Khaled, M. (2019). The finitely axomatizable complete theories of nonassociative arrow frames. Advances in Mathematics, 346(13), 194–218.CrossRefGoogle Scholar
Lyndon, R. C. (1961). Relation algebras and projective geometries. Michigan Mathematical Journal, 8, 21–28.Google Scholar
Maddux, R. D. (1981). Embedding modular lattices into relation algebras. Algebra Universalis, 12, 242–246.CrossRefGoogle Scholar
Maddux, R. D. (1991). The origin of relation algebras in the development and axiomatization of the calculus of relations. Studia Logica, 50, 421–455.CrossRefGoogle Scholar
Maddux, R. D. (2006). Relation Algebras. Amsterdam: North-Holland. xxvi+731 pp.Google Scholar
Maddux, R. D. (2018). Subcompletions of representable relation algebras. Algebra Universalis, 79(2), Paper 20, 32 pp.CrossRefGoogle Scholar
Marx, M. & Venema, Y. (1997). Multi-Dimensional Modal Logic. Dordrecht: Kluwer Academic Publishers. xiii+239 pp.CrossRefGoogle Scholar
McKenzie, R. (1970). Representations of integral relation algebras. Michigan Mathematical Journal, 17, 279–287.Google Scholar
Monk, J. D. (1964). On representable relation algebras. Michigan Mathematical Journal, 11, 207–210.Google Scholar
Pratt, V. R. (1990). Dynamic algebras as a well-behaved fragment of relation algebras. In Bergman, J., Maddux, R. D., and Pigozzi, D., editors. Algebraic Logic and Universal Algebra in Computer Science. Berlin: Springer-Verlag, pp. 77–110.CrossRefGoogle Scholar
Pratt, V. R. (1992). Origins of the calculus of binary relations. Logic in Computer Science, Proceedings of the 7th Annual IEEE Symposion, IEEE. pp. 248–254.Google Scholar
Schmidt, G. (2011). Relational Mathematics. Cambridge: Cambridge University Press. xiii+566 pp.Google Scholar
Simon, A. (1997). Nonrepresentable Algebras of Relations. Ph.D. Dissertation, Budapest: Hungarian Academy of Sciences. iii+86 pp.Google Scholar
Tarski, A. (1941). On the calculus of relations. The Journal of Symbolic Logic, 6, 73–89.CrossRefGoogle Scholar
Tarski, A. & Givant, S. (1987). A Formalization of Set Theory without Variables. Providence, Rhode Island: American Mathematical Society. xxi+318 pp.CrossRefGoogle Scholar