Abstract
It is well-known that in first-order logic, the theory of a binary relation and the theory of a ternary relation are mutually interpretable, i.e., each can be interpreted in the other. We establish the stronger result that they are interpretively isomorphic, i.e., they are mutually interpretable by a pair of interpretations each of which is the inverse of the other.
The author thanks Adam Gajdor, Wilfrid Hodges and Jan Mycielski for helpful comments and suggestions.
This is a preview of subscription content, log in via an institution.
Preview
Unable to display preview. Download preview PDF.
References
Barwise, J. A preservation theorem for interpretations. Cambridge Summer School in Mathematical Logic, Cambridge, 1971, Lecture Notes in Mathematics, Springer-Verlag, Berlin 337 (1973) 618–621
Bouvere, K.: Synonymous Theories. The Theory of Models (Addison, Henkin and Tarski, eds.), North-Holland, Amsterdam (1965) 402–406
Fause, D.: The Boolean algebra of formulas of first-order logic. Annals of Mathematical Logic 23 (1982) 27–53
Feferman, S.: Lectures on proof theory. Proceedings of the Summer School in Logic (Leeds, 1967), Lecture Notes in Mathematics, Berlin 70 (1968) 1–107
Gaifman, H.: Operations on relational structures, functors and classes. I. Proceedings of the Tarski Symposium, 1971, Proceedings of Symposia in Pure Mathematics 25, American Mathematical Society, Providence, R.I., (first edition: 1974; second edition: 1979) 21–39
Hanf, W.: Primitive Boolean algebras. Proceedings of the Alfred Tarski Symposium, Proceedings of Symposia in Pure Mathematics, American Mathematical Society, Providence, R.I. 25 (1974) 75–90
Hanf, W.: The Boolean algebra of logic. Bulletin of the American Mathematics Society 81 (1975) 587–589
Hanf, W., Myers, D.: Boolean sentence algebras: isomorphism constructions. Journal of Symbolic Logic 48 (1983) 329–338
Henkin, L., Mong, D., Tarski, A.: Cylindric Algebras. North-Holland, Amsterdam, 1971
Hodges, W.: A normal form for algebraic constructions II. Logique et Analyse 18 (1975) 429–487
Manders, K.: First-order logical systems and set-theoretic definability. preprint 1980
Montague, R.: Contributions to the Axiomatic Foundations of Set Theory. Ph.D. Thesis, University of California, 1957
Mycielski, J.: A lattice of interpretability types of theories. Journal of Symbolic Logic 42 (1977) 297–305
Mycielski, J., Pudlak, P., Stern, A.: A lattice of chapters of mathematics (interpretations between theories [theorems]. Memoirs of the American Mathematical Society 84 (1990) 1–70
Myers D.: Cylindric algebras of first-order theories. Transactions of the American Mathematical Society 216 (1976a) 189–202
Myers, D.: Invariant uniformization. Fundamenta Mathematicae 91 (1976b) 65–72
Myers, D.: Lindenbaum-Tarski algebras. Handbook of Boolean Algebras (Monk, ed.), Elsevier Science Pulishers, B.V. (1989) 1168–1195
Pillay, A.: Gaifman operations, minimal models and the number of countable models. Ph.D. Thesis, University of London, 1977
Prazmowski, K., Szczerba, L.: Interpretability and categoricity. Bulletin de l'Academie Polonaise des Sciences 24 (1976) 309–312
Rabin, M.: A simple method for undecidability proofs and some applications. Logic, Methodology and Philosophy of Science, Proc. 1964 International Congress, (Bar-Hillel, ed.), North-Holland, Amsterdam, (1965), 58–68
Simons, R.: The Boolean algebra of sentences of the theory of a function. Ph.D. Thesis, Berkeley, 1971
Szczerba, L.: Interpretability of elementary theories. Logic, Foundations of Mathematics and Computability Theory (Butts and Hintikka, eds.), D. Reidel Publishing Co., Dordrecht-Holland (1977) 126–145
Szmielew, W., Tarski, A.: Mutual interpretability of some essentially undecidable theories. Proceedings of the International Congress of Mathematicians, Cambridge, Massachusetts, American Mathematical Society, Providence (1952) 734
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1997 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Myers, D. (1997). An interpretive isomorphism between binary and ternary relations. In: Mycielski, J., Rozenberg, G., Salomaa, A. (eds) Structures in Logic and Computer Science. Lecture Notes in Computer Science, vol 1261. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-63246-8_6
Download citation
DOI: https://doi.org/10.1007/3-540-63246-8_6
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-63246-7
Online ISBN: 978-3-540-69242-3
eBook Packages: Springer Book Archive