Hostname: page-component-7b7567c8c-tnzcf Total loading time: 0 Render date: 2026-11-14T06:18:16.215Z Has data issue: false hasContentIssue false
  • English
  • Français

Atomless varieties

Published online by Cambridge University Press:  12 March 2014

Yde Venema
Yde Venema*
Affiliation:
Institute of Logic, Language and Computation, University of Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands, E-mail: yde@science.uva.nl
Get accessRights & Permissions [Opens in a new window]

Abstract

We define a nontrivial variety of boolean algebras with operators such that every member of the variety is atomless. This shows that not every variety of boolean algebras with operators is generated by its atomic members, and thus establishes a strong incompleteness result in (multi-)modal logic.

Information

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 68 , Issue 2 , June 2003, pp. 607 - 614
Copyright
Copyright © Association for Symbolic Logic 2003

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.)

Article purchase

Temporarily unavailable

References

REFERENCES

[1]Blackburn, P., de Rijke, M., and Venema, Y., Modal Logic, Cambridge University Press, 2001.CrossRefGoogle Scholar
[2]Givant, S., Universal classes of simple relation algebras, this Journal, vol. 64 (1999), pp. 575–589.Google Scholar
[3]Goldblatt, R., Varieties of Complex Algebras, Annals of Pure and Applied Logic, vol. 38 (1989), pp. 173–241.CrossRefGoogle Scholar
[4]Goldblatt, R., Persistence and atomic generation for varieties of Boolean algebras with operators, Studia Logica, vol. 68 (2001), pp. 155–171.CrossRefGoogle Scholar
[5]Jipsen, P., Discriminator varieties of Boolean algebras with residuated operators. In Rauszer [8], pp. 239–252.CrossRefGoogle Scholar
[6]Kracht, M., de Rijke, M., Wansing, H., and Zakharyaschev, M. (editors), Advances in Modal Logic, Volume 1, CSLI Publications, 1998.Google Scholar
[7]Kracht, M. and Kowalski, T., Atomic incompleteness or how to kill one bird with two stones, Bulletin of the Section of Logic, vol. 30 (2001), pp. 71–78.Google Scholar
[8]Rauszer, C. (editor), Algebraic Methods in Logic and Computer Science, Banach Center Publications, vol. 28, Polish Academy of Sciences, 1999.Google Scholar
[9]Thomason, S. K., An incompleteness theorem in modal logic, Theoria, vol. 40 (1974), pp. 150–158.CrossRefGoogle Scholar
[10]Venema, Y., Atom structures, In Kracht et al. [6], pp. 291–305.Google Scholar