Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-25T07:19:59.495Z Has data issue: false hasContentIssue false
  • English
  • Français

Undecidable semiassociative relation algebras

Published online by Cambridge University Press:  12 March 2014

Roger D. Maddux
Roger D. Maddux*
Affiliation:
Department of Mathematics, Iowa State University, Ames, Iowa 50011-2066, E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

If K is a class of semiassociative relation algebras and K contains the relation algebra of all binary relations on a denumerable set, then the word problem for the free algebra over K on one generator is unsolvable. This result implies that the set of sentences which are provable in the formalism ℒw× is an undecidable theory. A stronger algebraic result shows that the set of logically valid sentences in ℒw× forms a hereditarily undecidable theory in ℒw×. These results generalize similar theorems, due to Tarski, concerning relation algebras and the formalism ℒ×.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 59 , Issue 2 , June 1994 , pp. 398 - 418
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

[CT51] Chin, L. H. and Tarski, A., Distributive and modular laws in the arithmetic of relation algebras, University of California Publications in Mathematics, New Series, vol. 1 (1951), pp. 341–384.Google Scholar
[D58] Davis, M., Computability and unsolvability, McGraw-Hill, New York, 1958.Google Scholar
[J82] Jónsson, B., Varieties of relation algebras, Algebra Universalis, vol. 15 (1982), pp. 273–298.Google Scholar
[JT52] Jónsson, B. and Tarski, A., Boolean algebras with operators, part II, American Journal of Mathematics, vol. 74 (1952), pp. 127–162.Google Scholar
[MMT87] McKenzie, R. N., McNulty, G. F., and Taylor, W. F., Algebras, lattices, and varieties, Volume I, Wadsworth and Brooks/Cole, Monterey, California, 1987.Google Scholar
[M78] Maddux, R. D., Topics in relation algebras, Doctoral dissertation, University of California, Berkeley, California, 1978.Google Scholar
[M80] Maddux, R. D., The equational theory of CA3 is undecidable, this Journal, vol. 145 (1980), pp. 311–316.Google Scholar
[M82] Maddux, R. D., Some varieties containing relation algebras, Transactions of the American Mathematical Society, vol. 272 (1982), pp. 501–526.Google Scholar
[M91] Maddux, R. D., Pair-dense relation algebras, Transactions of the American Mathematical Society, vol. 328 (1991), pp. 83–131.Google Scholar
[N86] Németi, I., Free algebras and decidability in algebraic logic, Doctoral dissertation, Hungarian Academy of Sciences, Budapest, 1986.Google Scholar
[N87] Németi, I., Decidability of relation algebras with weakened associativity, Proceedings of the American Mathematical Society, vol. 100 (1987), pp. 340–344.Google Scholar
[TG87] Tarski, A. and Givant, G., A formalization of set theory without variables, Colloquium Publications, vol. 41, American Mathematical Society, Providence, Rhode Island, 1987.Google Scholar